Making Waves

Abstract

We study linear PDE constraints for vector-valued functions and distributions. Our focus lies on wave solutions, which give rise to distributions with low-dimensional support. Special waves from vector potentials are represented by syzygies. We parametrize all waves by projective varieties derived from the support of the PDE. These include determinantal varieties and Fano varieties, and they generalize wave cones in analysis.

1 Introduction

We consider homogeneous, linear systems of partial differential equations with constant coefficients. In this article, the acronym PDE is used to refer to such systems. Fusing notational conventions from [1, 2] and [3, 4], we study PDE for unknown functions $\phi :\mathbb {R}^n \rightarrow \mathbb {C} ^k$. If the system has $\ell$ constraints then we write the PDE as an $\ell \times k$ matrix $A = (a_{ij})$ with entries in the polynomial ring $R=\mathbb {C} [\partial _1,\dotsc ,\partial _n]$. We assume that all entries $a_{ij}$ are homogeneous of the same degree d, so d is the order of our PDE. The action of the matrix A on $\phi = \phi (x)$ results in a column vector $A \bullet \phi$ of length $\ell$, whose ith entry equals $\sum _{j=1}^k a_{ij} \bullet \phi _j$, where $a_{ij}(\partial )$ acts on $\phi (x)$ via $\partial _s = \frac{\partial }{\partial x_s}$. The $\ell$ rows of A generate a submodule $M = {{\,\textrm{Im}\,}}_R(A^T)$ of the free module $R^k$.

The solutions $\phi$ to A are drawn from appropriate spaces of distributions [5], and depend only on the module M. The spaces we need will be reviewed in Sect. 2. The support of M is the variety defined by the $k \times k$ minors of A. We are interested in cases these minors vanish identically, i.e., where the matrix A has rank less than k, for instance if $\ell < k$.

Example 1.1

($n=3, k=4, \ell =2$) We illustrate our setup for the PDE

$$\begin{aligned} A \quad = \quad \begin{bmatrix} \partial _1 &{} \partial _2 &{} \partial _3 &{} 0 \\ 0 &{} \partial _1 &{} \partial _2 &{} \partial _3 \end{bmatrix}. \end{aligned}$$

(1)

Solutions to A are functions $\phi : \mathbb {R}^3 \rightarrow \mathbb {C} ^4$ that satisfy $A \bullet \phi = 0$. In coordinates, this reads

$$\begin{aligned} \frac{\partial \phi _1}{\partial x_1} + \frac{\partial \phi _2}{\partial x_2} + \frac{\partial \phi _3}{\partial x_3} \,\,= \,\, \frac{\partial \phi _2}{\partial x_1} + \frac{\partial \phi _3}{\partial x_2} + \frac{\partial \phi _4}{\partial x_3} \,\,= \,\, 0. \end{aligned}$$

To find some solutions, we write the kernel of A as the image of the skew-symmetric matrix

For any function $\psi :\mathbb {R}^3 \rightarrow \mathbb {C} ^4$ we get a solution $\phi = B\bullet \psi$, since $A \bullet (B \bullet \psi ) = (A B) \bullet \psi = 0$. We call $\psi$ a vector potential. For instance, if $\varphi$ is any scalar function and $\psi = \varphi \cdot e_1$, then

(2)

The matrix B has rank two, so its column span over the field $K = \mathbb {C} (\partial _1,\partial _2,\partial _3)$ has a basis of two vectors in $K^4$. For readers of [1] we note that $M = {{\,\textrm{Im}\,}}_R(A^T)$ is a $\{0 \}$-primary submodule of multiplicity 2 in $R^4$. The Macaulay2 command solvePDE outputs two basis vectors. $\triangle$

This project arose from our desire to understand the hierarchy of wave cones in [3]. These are subsets of $\mathbb {C} ^k$ which play an important role in the regularity theory of PDEs. Recent work on the algebraic side in [2, 6] pioneered differential primary decompositions, where modules M are encoded by their associated primes along with certain vectors of length k, known as Noetherian operators. Our presentation connects these threads from analysis and algebra. Along the way, we give an exposition of known results from control theory [7,8,9,10].

The waves in our catchy title are special solutions to the PDE. We develop geometric theory and algebraic algorithms for finding them. Our point of departure is the simple wave

$$\begin{aligned} \phi _{\xi ,u}(x) \,\,\,= \,\,\, \textrm{exp}(i \xi \cdot x) \cdot u. \end{aligned}$$

(3)

Here, $\xi \in \mathbb {R}^n$, $u \in \mathbb {C} ^k$, and $i = \sqrt{-1}$. The exponential function is applied to the dot product of $x = (x_1,\ldots ,x_n)$ with the purely imaginary vector $i \xi$, resulting in trigonometric functions. The real vector $\xi$ is the frequency, while the complex vector u is the amplitude. Simple wave solutions are characterized by a system of $\ell$ polynomial equations in their $n+k$ coordinates:

$$\begin{aligned} A \bullet \phi _{\xi ,u} = 0 \quad \hbox {if and only if} \quad A(\xi ) \cdot u = 0. \end{aligned}$$

(4)

For the PDE in Example 1.1, setting $\varphi (x) = \textrm{exp}( i \xi \cdot x)$ in (2) yields a simple wave solution.

Our standing assumption is that all entries of the matrix A are homogeneous polynomials in $\partial _1,\ldots ,\partial _n$ of the same degree d. This implies that A is elliptic—therefore smoothing—if and only if there are no nontrivial wave solutions. Thus, the existence of wave solutions has a major impact on the analytical properties of the operator, cf. [11, Chapter 2.1].

Wave solutions are obtained from superpositions of simple waves and taking limits. In the superpositions we allow here, the amplitude u is fixed, whereas the frequency $\xi$ runs over linear subspaces of $\mathbb {R}^n$ all of whose points satisfy (4). Taking limits of such superpositions leads to waves that are distributions with low-dimensional support. This construction will be explained in detail in Sect. 2, where we also review basics on spaces of functions and distributions.

In Sect. 3 we construct solutions that arise from a vector potential. If M is $\{0 \}$-primary then all solutions to A have that property. This is based on results in commutative algebra and control theory, notably due to Shankar [9, 10]. The vector potential can be used to build distributional solutions that are compactly supported and act like waves in the interior of their support.

In Sect. 4 we turn to algebraic geometry, and we introduce projective varieties that parametrize wave solutions. These generalize the determinantal varieties of matrices of linear forms. In Sect. 5 we examine the analytic meaning of wave varieties and obstruction varieties, and discuss the analytic implications of working algebraically in complex projective spaces. In Sect. 6 we introduce varieties of wave pairs, which generalize Fano varieties [12, Example 6.19] inside Grassmannians. We present methods for computing wave pairs and wave varieties, and we illustrate these on several examples. In the context of a given PDE A, these scenarios give interesting distributional solutions to A with low-dimensional support.

We close the introduction with a well-known equation from the theory of elasticity [13, 14].

Example 1.2

(Saint-Venant’s tensor) Set $d=2,k = \left( {\begin{array}{c}n+1\\ 2\end{array}}\right) ,\ell = k^2$, and identify $\mathbb {C} ^k$ with the space of symmetric $n \times n$ matrices. We consider matrix-valued distributions $\phi : \mathbb {R}^n \rightarrow \mathbb {C} ^k$. The Saint-Venant operator A characterizes the kernel of the 2-dimensional X-ray transform:

$$\begin{aligned} A \bullet \phi \,\,=\,\, \bigl ( \partial _i\partial _j \phi _{ab} + \partial _a\partial _b \phi _{ij} - \partial _i\partial _a \phi _{jb} - \partial _j \partial _b \phi _{ia} \bigr )_{i,j,a,b = 1,\dotsc ,n}. \end{aligned}$$

(5)

In our notation, A is an $\ell \times k$ matrix whose nonzero entries are quadratic monomials $\partial _i \partial _j$. By removing redundant rows, using [13], the number of rows of A can be reduced to $\ell = \frac{1}{6} \left( {\begin{array}{c}n^2\\ 2\end{array}}\right)$. The PDE A has a vector potential B, as in Sect. 3, given by the symmetric gradient:

$$\begin{aligned} B \bullet \psi \,\,=\,\, \bigl ( \, \partial _i \psi _j + \partial _j \psi _i\,\bigr )_{i,j = 1,\dotsc ,n}. \end{aligned}$$

The wave pair variety ${\mathcal {P}}_A^{n-1}$ of Sect. 6 lives in the space $\mathbb {P}^{n-1} \times \mathbb {P}^{k-1}$, and it coincides with the incidence variety ${\mathcal {I}}_A$ in (20). It is defined by the $3 \times 3$ minors of the $(n+1) \times (n+1)$-matrix

$$\begin{aligned} \begin{bmatrix} 0 &{} y_1 &{} y_2 &{} \cdots &{} y_n \\ y_1 &{} z_{11} &{} z_{12} &{} \cdots &{} z_{1n} \\ y_2 &{} z_{12} &{} z_{22} &{} \cdots &{} z_{2n} \\ \vdots &{} \vdots &{} \vdots &{} \ddots &{} \vdots \\ y_n &{} z_{1n} &{} z_{2n} &{} \cdots &{} z_{nn} \end{bmatrix}. \end{aligned}$$

(6)

The wave variety ${\mathcal {W}}_A \subset \mathbb {P}^{k-1}$ of Sect. 4 is given by the $3 \times 3$ minors of the $n \times n$ matrix $(z_{ij})$. This example is a variant of the curl operator in Proposition 6.6, with (6) playing the role of (30), and underscores the relevance of nonlinear algebra [15] for the physical sciences. $\triangle$

2 Spaces and Waves

In real analysis it is customary to consider functions from the real space $\mathbb {R}^n$ to the complex space $\mathbb {C} ^k$. By passing to the real part or the imaginary part, one obtains a function from $\mathbb {R}^{n}$ to $\mathbb {R}^{k}$. For instance, the real part of the simple wave (3) is an expression in terms of sine and cosine. If $\phi$ satisfies a linear PDE A with real coefficients then its real part satisfies A.

We now fix the setting found in standard analysis texts, such as Hörmander’s book [5]. Fix an open convex subset $\Omega \subset \mathbb {R}^n$. Let $C_{c}^\infty (\Omega ,\mathbb {C} ^{k})$ denote the space of smooth functions $f :\mathbb {R}^n \rightarrow \mathbb {C} ^{k}$ whose support is compact and contained in $\Omega$. One writes ${\mathcal {D}} = C_{c}^{\infty (\Omega ,\mathbb {C} ^{k})}$ for the same space after endowing it with the topology of sequential convergence. Here, $\{f_k\}$ converges to f in ${\mathcal {D}}$ if there exists a compact set $K \subset \Omega$, containing the supports of f and of each $f_k$, such that the norm of $\,\partial ^{\alpha } f_{k - \partial }^{\alpha } f\,$ converges to zero for each $\alpha \in \mathbb {N}^{n}$. Thus, ${\mathcal {D}}$ is a topological vector space over $\mathbb {C}$ which is a module over $R = \mathbb {C} [\partial _1,\ldots ,\partial _n]$ by differentiation.

The space of distributions ${\mathcal {D}}'$ is a subspace of the vector space dual to ${\mathcal {D}}$. Its elements are the linear functions ${\mathcal {D}} \rightarrow \mathbb {C}$ that are continuous in the topology above. This can be expressed by the following growth condition: a distribution is a linear functional $\phi :{\mathcal {D}} \rightarrow {\mathbb {C}}$ such that, for every compact set $K \subset {\mathbb {R}}^n$, there exist positive real constants C and d with

We denote by ${\mathcal {D}}'$ the space of distributions from $\Omega$ to $\mathbb {C} ^{k}$. Every compactly supported smooth function is also a distribution, as there is a natural inclusion from ${\mathcal {D}}$ into $\mathcal {D'}$ which takes a function $\phi : \mathbb {R}^{n} \rightarrow \mathbb {C} ^{k}$ to the linear functional ${\mathcal {D}} \rightarrow \mathbb {C} , f \mapsto \int f \cdot \phi$ for all $f \in {\mathcal {D}}$. Other prominent examples of distributions are the Dirac delta functions and the step functions.

We now come to the special role played by exponential functions. For this, we introduce the Schwartz space ${\mathcal {S}} = {\mathcal {S}}(\mathbb {R}^n,\mathbb {C} ^k)$ whose elements are smooth functions f such that $||x^\beta \partial ^\alpha f||_\infty$ is finite for all $\alpha ,\beta \in \mathbb {N}^n$. This space includes the simple waves (3), since the coordinates of $\xi$ are real. However, many nice functions, such as polynomials, are not in ${\mathcal {S}}$.

Most relevant for us is that ${\mathcal {D}} = C^\infty _c(\Omega ,\mathbb {C} ^k)$ is a subspace of ${\mathcal {S}} = {\mathcal {S}}(\mathbb {R}^n,\mathbb {C} ^k)$. The key feature of the Schwartz space is the endomorphism known as the Fourier transform $\,\hat{}: {\mathcal {S}} \rightarrow {\mathcal {S}}$. By applying $\,\,\hat{}\,\,$ twice, we see that every function in ${\mathcal {S}}$ admits an integral representation

$$\begin{aligned} f(x) \,\,\, = \,\,\, \int _{\mathbb {R}^n} \! \textrm{exp}(\,2 \pi i \,\xi \cdot x)\, {{\hat{f}}}(\xi ) \,d \xi . \end{aligned}$$

(7)

The dual to the Schwarz space consists of the tempered distributions, and we have inclusions

$$\begin{aligned} {\mathcal {D}}\, \hookrightarrow \,{\mathcal {S}} \,\hookrightarrow \, {\mathcal {S}}' \,\hookrightarrow \, {\mathcal {D}}'. \end{aligned}$$

(8)

All of these spaces are R-modules because the linear map $\partial ^\alpha : {\mathcal {D}} \rightarrow {\mathcal {D}}$ is continuous, so we get a dual $(\partial ^\alpha )^*: {\mathcal {D}}' \rightarrow {\mathcal {D}}'$ which restricts to ${\mathcal {S}}'$ and ${\mathcal {S}}$. One subtle issue here is the sign, as the action of $R = \mathbb {C} [\partial _1,\ldots ,\partial _n]$ on distributions by taking partial derivatives satisfies

$$\begin{aligned} (\partial _i \bullet \phi ) (f)\,\, = \,\,-\phi \bigl ( {\partial f/ \partial x_i}\bigr ). \end{aligned}$$

The notion of waves used in this paper arises from superpositions of the simple waves (3),

$$\begin{aligned} \phi (x) \,\,=\,\, \sum _{j=1}^p \lambda _j \phi _{\xi _j,u} (x). \end{aligned}$$

(9)

Here, the amplitude $u \in \mathbb {C} ^k$ is fixed but the frequencies $\xi _1,\ldots ,\xi _p$ vary in $\mathbb {R}^n$. The coefficients $\lambda _1,\ldots ,\lambda _p$ are complex numbers. If each summand in (9) satisfies A then so does $\phi (x)$. The integral representation (7) of Schwartz functions by the Fourier transform implies that every distribution $\delta \in {\mathcal {D}}'$ can be approximated by a sequence of waves $\phi ^{(1)}, \phi ^{(2)}, \ldots$ of the form (9).

Lemma 2.1

The linear span of the exponential functions $x \mapsto \textrm{exp}(i\xi \cdot x)$ is dense in ${\mathcal {D}}'$.

Our aim is to create interesting distributions by taking limits of waves (9) in ${\mathcal {D}}'$. To this end, suppose that $\xi _1,\ldots ,\xi _p$ span a linear subspace $\pi$ of $\mathbb {R}^n$ such that $A \bullet \phi _{\xi ,u} = 0$ for all $\xi \in \pi$. We then consider the closure in ${\mathcal {D}}'$ of the space of all waves (9) whose frequencies $\xi _j$ are in $\pi$, i.e., a set of distributional solutions with low-dimensional support satisfying the PDE A. This motivates the following definitions. As before, $A \in R^{\ell \times k}$ is a matrix whose entries are homogeneous of degree d. A wave pair for A is a pair $(u, \pi )$, where $u \in \mathbb {C} ^k$ and $\pi$ is a linear subspace of $\mathbb {R}^n$, such that $A(\xi ) u = 0$ for all $\xi \in \pi$. If $(u,\pi )$ is a wave pair then any superposition (9) with $\xi _1,\ldots ,\xi _p \in \pi$ is a classical wave solution of A. A wave solution to A is any distribution in the closure in ${\mathcal {D}}'$ of the classical wave solutions.

Proposition 2.2

Consider any wave pair $(u,\pi )$ for the operator A and set $r = \textrm{codim}(\pi )$. The associated wave solutions are precisely the distributions of the form

$$\begin{aligned} \qquad \phi (x) \,=\,\delta \bigl (L_1(x),\ldots ,L_{n-r}(x) \bigr ) \cdot u, \end{aligned}$$

(10)

where $L_1, \dotsc , L_{n-r}$ are linear form satisfying $\pi ^\perp = \{x \in \mathbb {R}^n: L_1(x) = \cdots = L_{n-r}(x) = 0\}$ and $\delta$ is any distribution in ${\mathcal {D}}'(\mathbb {R}^{n-r},\mathbb {C} )$. Thus, equation (10) characterizes wave pairs as follows: if $\phi (x)$ is a solution to the PDE A for all $\delta \in {\mathcal {D}}'(\mathbb {R}^{n-r},\mathbb {C} )$ then $(u, \pi )$ is a wave pair.

Remark 2.3

The notation $\delta (L\cdot ):= \delta (L_1(x), \dotsc , L_{n-r}(x))$ refers to an extension from smooth functions to distributions. Following [5, Chapter 6], one can define it as follows. Given a real matrix $L \in \mathbb {R}^{(n-r)\times r}$, fix an orthonormal basis $v_1,\dotsc , v_r$ for $\ker (L)$ and let $L'\in \mathbb {R}^{r\times n}$ be the matrix with the $v_i$ as rows. The matrix $H = \begin{bmatrix} L \\ L' \end{bmatrix}$ defines an endomorphism $\mathbb {R}^n \rightarrow \mathbb {R}^n,\,x \mapsto (y',y'')$, where $y' \in \mathbb {R}^{n-r}$, $y'' \in \mathbb {R}^r$. Its inverse is $H^{-1} = \begin{bmatrix} L^T(LL^T)^{-1}&L'^T \end{bmatrix}$. If $\delta :\mathbb {R}^{n-r} \rightarrow \mathbb {C}$ is smooth, then, by a change of variables, for any test function $f \in {\mathcal {D}}(\mathbb {R}^n, \mathbb {C} )$,

We write this as $\, \delta (L\cdot )(f) = \delta (1(F))$, with the constant function $1: \mathbb {R}^r \rightarrow \mathbb {C}$ and $F(y) = \frac{1}{\sqrt{\det (LL^T)}}f(H^{-1}y)$. There exists a unique distribution $\delta \otimes 1$ such that $(\delta \otimes 1)(F) = \delta (1(F)) = 1(\delta (F))$ for all $F \in {\mathcal {D}}(\mathbb {R}^n,\mathbb {C} )$. Now define $\delta (L\cdot )(f):= (\delta \otimes 1)(F)$ for arbitrary distributions $\delta$.

Proof of Proposition 2.2

Write $x = (x_1,\dotsc ,x_n)$ and $y = (y_1,\dotsc ,y_{n-r})$ for the coordinates of $\mathbb {R}^n$ and $\mathbb {R}^{n-r}$, and let L denote the $(n-r) \times n$ matrix of coefficients of $L_1,\dotsc , L_{n-r}$. For $\eta \in \mathbb {R}^{n-r}$, consider the wave function $x \mapsto \delta _\eta (Lx)\cdot u$ associated with the exponential function $\delta _\eta (y) = \exp (i\eta \cdot y)$. Applying the differential operator A to that wave function yields

$$\begin{aligned} A \bullet (\delta _\eta ( Lx) \cdot u) \, = \, A \bullet ( \textrm{exp}(i\eta Lx) \cdot u) \,=\,i^d\, \textrm{exp}(i\eta L x) \cdot A(\eta L) u. \end{aligned}$$

(11)

This vector of length $\ell$ is zero for all $\eta \in \mathbb {R}^{n-r}$ if and only if $(u, \pi )$ is a wave pair. Since the space spanned by the exponential functions $\delta _\eta$ for $\eta \in \mathbb {R}^{n-r}$ is dense in the space of all distributions, by Lemma 2.1, the first assertion follows.

The second statement follows from the fact that $A\cdot (\delta _\eta (Lx)\cdot u)=0$ for all $\eta$ if and only if $(u,\pi )$ is a wave pair, together with the simple observation that $\delta _\eta \in {\mathcal {D}}'(\mathbb {R}^{n-r},\mathbb {C} )$. $\square$

We seek wave pairs $(u,\pi )$, where r is as small as possible, as this enables the construction of distributional solutions with small support. What follows is the standard construction.

Remark 2.4

Let $\delta$ be the Dirac delta distribution at the origin in $\mathbb {R}^{n-r}$. The distribution $\phi$ in (10) is supported on the r-dimensional subspace $\pi ^\perp$ of $\mathbb {R}^n$, and it satisfies the PDE A if $(u, \pi )$ is a wave pair. Such A-free measures are important in the calculus of variations [3, 4].

Example 2.5

($n=3, k=4, r=2$) Fix the matrix A in Example 1.1. For all $\xi \in \mathbb {C} ^3 \backslash \{0\}$, the linear space $\,\textrm{ker} \,A(\xi )$ has dimension 2. It consists of all vectors $u \in \mathbb {C} ^4$ such that

$$\begin{aligned} \begin{bmatrix} \xi _1 &{} \xi _2 &{} \xi _3 &{} 0 \\ 0 &{} \xi _1 &{} \xi _2 &{} \xi _3 \end{bmatrix} \begin{bmatrix} u_1 \\ u_2 \\ u_3 \\ u_4 \end{bmatrix} \,\, = \,\, \begin{bmatrix} u_1 &{} u_2 &{} u_3 \\ u_2 &{} u_3 &{} u_4 \end{bmatrix} \begin{bmatrix} \xi _1 \\ \xi _2 \\ \xi _3 \end{bmatrix} \,\, = \,\, \begin{bmatrix} 0 \\ 0 \end{bmatrix}. \end{aligned}$$

(12)

This equation characterizes simple waves $\phi _{\xi ,u}$ that satisfy A. With $r=2$ in (10) we can take

$$\begin{aligned} L_1(x) \,\,=\,\, (u_3^2-u_2 u_4)x_1 + (u_1u_4-u_2 u_3) x_2 + (u_2^2-u_1 u_3) x_3 \quad \hbox {for any} u \in \mathbb {R}^4.\nonumber \\ \end{aligned}$$

(13)

By superposition we obtain waves with $r=1$. Here, u must be chosen such that the three coefficients in (13) vanish. This means that u lies in the cone over the twisted cubic curve:

$$\begin{aligned} (u_1,u_2,u_3,u_4) \,\, = \,\, (s^3,s^2t, st^2,t^3). \end{aligned}$$

(14)

This is the wave variety ${\mathcal {W}}^1_A \subset \mathbb {P}^3$ in Example 4.3. We obtain wave pairs $(u,\pi )$ with $\textrm{codim}(\pi ) = 2$, and thus solutions supported on a plane in $\mathbb {R}^4$, by taking the two linear forms

$$\begin{aligned} L_1(x) \,=\, t x_1 - s x_2 \quad \textrm{and} \quad L_2(x) \,=\, t x_2 - s x_3. \end{aligned}$$

Indeed, $\phi (x) = \delta \bigl (L_1(x),L_2(x) \bigr ) \cdot u$ is a wave solution of A, for any $\delta \in {\mathcal {D}}'(\mathbb {R}^2,\mathbb {C} )$. $\triangle$

3 Solutions from syzygies

We now focus on solutions to the PDE $A\bullet \phi$ that are represented by a vector potential $\psi$ as in Example 1.1. This section is independent from the rest of the paper. It offers tools for constructing compactly supported solutions with desirable properties, including those that are waves in the interior of their support. From an analytic point of view, the existence of compactly supported solutions is of interest in the context of understanding the space of Young measures; see e.g., [16]. We note that, while the functions in ${\mathcal {D}}$ are not waves, we can fuse them with waves to create solutions that are of interest in fields such as convex integration [17, 18]. The choice of the potential $\psi$ allows for this.

Let ${\mathcal {F}}$ be a space of functions or distributions $\mathbb {R}^n \rightarrow \mathbb {C}$ such as those in (8), or those in [7,8,9,10]. We assume that ${\mathcal {F}}$ is an R-module under differentiation. Solving the PDE means describing all k-tuples $\phi \in {\mathcal {F}}^k$ with $A\bullet \phi = 0$. The set of all solutions is the R-module

$$\begin{aligned} {{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(A) \,:= \, {{\,\textrm{Ker}\,}}_{{\mathcal {F}}}(A) \,\,\subset \,\, {\mathcal {F}}^k. \end{aligned}$$

Algebraic algorithms for this task, along with implementations in Macaulay2, are presented in [1, 19]. We note that $\phi \in {{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(A)$ if and only if $m \bullet \phi = 0$ for all $m \in M$, where M is the submodule of $R^k$ spanned by the rows of A. Thus the solution space depends only on the rowspan of A. More formally, we apply the functor ${{\,\textrm{Hom}\,}}_R(\,\, \cdot \,\,, {\mathcal {F}})$ to the exact sequence

$$\begin{aligned} R^\ell \,\xrightarrow {A^T} \,R^k \,\rightarrow \, R^k/M \,\rightarrow \, 0. \end{aligned}$$

The result of this dualization step is the sequence ${\mathcal {F}}^\ell \,\xleftarrow {A} \,{\mathcal {F}}^k \,\leftarrow \, {{\,\textrm{Hom}\,}}_R(R^k/M, {\mathcal {F}})\, \leftarrow \, 0$. This sequence is also exact, so our solution space can be written as an R-module as follows:

$$\begin{aligned} {{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(A) \,=\, {{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(M) \,\cong \,{{\,\textrm{Hom}\,}}_R(R^k/M, {\mathcal {F}}). \end{aligned}$$

(15)

Note that ${{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(\,\, \cdot \,\,)$ is inclusion reversing: if $M_0 \subseteq M_1$ are submodules of $R^k$, then ${{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(M_0) \supseteq {{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(M_1)$. To question to what extent a module M of PDEs can be recovered from its solution spaces is addressed by the Nullstellensatz in [9]. Indeed, there is a considerable body of literature in control theory on the connection between linear PDE and commutative algebra. We here follow the expositions by Oberst [7, 8] and Shankar [10], and the references therein.

Recall that an R-module ${\mathcal {F}}$ is injective if the functor ${{\,\textrm{Hom}\,}}(\,\, \cdot \,, {\mathcal {F}})$ is exact. An R-module ${\mathcal {F}}$ is an injective cogenerator (cf. [7]) if the following holds: a sequence of R-modules $A \rightarrow B \rightarrow C$ is exact if and only if ${{\,\textrm{Hom}\,}}_R(A, {\mathcal {F}}) \leftarrow {{\,\textrm{Hom}\,}}_R(B, {\mathcal {F}}) \leftarrow {{\,\textrm{Hom}\,}}_R(C,{\mathcal {F}})$ is exact. The following result concerns the space ${\mathcal {D}}'$ of distributions. It fails for the subspace ${\mathcal {D}}$ of ${\mathcal {D}}'$.

Proposition 3.1

[7, Corollary 4.36] The R-module ${\mathcal {D}}'$ is an injective cogenerator, so the inclusion reversing map from submodules $M \subseteq R^k$ to solution spaces ${{\,\textrm{Sol}\,}}_{{\mathcal {D}}'}(M)$ is bijective.

Our goal is to compute the subspace of solutions to A that are derived from vector potentials as in (2). This is usually a proper subspace, as seen in the following simple example.

Example 3.2

($k=\ell =n=d=2$) Let M be the R-module generated by the rows of

$$\begin{aligned} A \,=\, \begin{bmatrix} \partial _1^2 &{} \partial _1 \partial _2 \\ \partial _1 \partial _2 &{} \partial _2^2 \end{bmatrix}. \end{aligned}$$

The solutions come in two flavors, corresponding to a primary decomposition $M = M_0 \,\cap \,M_1$. Namely, $\textrm{Sol}_{\mathcal {F}}(M) = \textrm{Sol}_{\mathcal {F}}(M_0) + \textrm{Sol}_{\mathcal {F}}(M_1)$, where $M_i$ is the module generated by the rows of

$$\begin{aligned} A_0 \,=\, \begin{bmatrix} \partial _1&\partial _2 \end{bmatrix} \quad \textrm{and} \quad A_1 \,=\, \begin{bmatrix} \partial _1^2 &{} \partial _1\partial _2 &{} \partial _2^2 &{} 0 &{} 0 &{} 0\\ 0 &{} 0 &{} 0 &{} \partial _1^2 &{} \partial _1\partial _2 &{} \partial _2^2 \end{bmatrix}^T. \end{aligned}$$

The solutions to the PDE $A_0$ are $\left[ \begin{array}{l} - \partial _2 \bullet \psi \\ \quad \partial _1 \bullet \psi \end{array}\right]$ for any $\psi \in {\mathcal {F}}$, while the solutions to $A_1$ are $\begin{bmatrix} ax_1+bx_2+c \\ a' x_1 + b'x_2 + c' \end{bmatrix}$ for $a,a',b,b',c,c' \in \mathbb {C}$. Both the minimal prime $\{0\}$ and the embedded prime $\langle \partial _1,\partial _2 \rangle$ have multiplicity one in M. We can assume $b=c=a'=b'=c'=0$, after a suitable choice of $\psi$. Such a minimal representation of the solution space can also be seen from the output of solvePDE in Macaulay2 [1, 19, 20]. $\triangle$

To achieve our goal for an arbitrary matrix $A \in R^{\ell \times k}$, we compute a matrix B such that

$$\begin{aligned} R^{k'} \xrightarrow {B} R^k \xrightarrow {A} R^\ell \end{aligned}$$

(16)

is an exact sequence. The columns of B are syzygies of A. The transpose of this sequence is

$$\begin{aligned} R^{k'} \xleftarrow {B^T} R^k \xleftarrow {A^T} R^\ell . \end{aligned}$$

(17)

This is a complex but it is generally not exact. Applying ${{\,\textrm{Hom}\,}}_R(\,\, \cdot \,, {\mathcal {F}})$, we get the complex

$$\begin{aligned} {\mathcal {F}}^{k'} \xrightarrow {B} {\mathcal {F}}^{k} \xrightarrow {A} {\mathcal {F}}^\ell , \end{aligned}$$

(18)

and hence ${{\,\textrm{Im}\,}}_{\mathcal {F}} (B) \subseteq {{\,\textrm{Sol}\,}}_{\mathcal {F}}(A)$. This means that $B \bullet \psi$ is a solution to our PDE A for any $\psi \in {\mathcal {F}}$. If the equality ${{\,\textrm{Im}\,}}_{\mathcal {F}} (B) = {{\,\textrm{Sol}\,}}_{\mathcal {F}}(A)$ holds then we say that A admits a vector potential. This was the case in Example 1.1 but not in Example 3.2, where (17) is not exact.

We briefly recall some definitions. An element f in an R-module U is a torsion element if $rf = 0$ for some $r \in R \backslash \{0\}$. The torsion submodule of U is the module of torsion elements. The module U is torsion if it is equal to its torsion submodule, and is torsion-free if its torsion submodule is zero. A prime ideal $P \subset R$ is associated to U if there is some element $u \in U$ that $P:= \{r \in R :ru =0 \}$. The set of associated primes of U is denoted $\mathop {\textrm{Ass}}\limits (U)$, and the module U is called P-primary if $\mathop {\textrm{Ass}}\limits (U) = \{P\}$. By standard abuse of language, we use the adjectives torsion, torsion-free, associated, and primary to describe properties of a submodule $M \subset R^k$ when $U = R^k/M$ has that property.

Theorem 3.3

Suppose that the sequence (16) is exact. Then the following are equivalent:

• (1) The PDE A admits a vector potential, i.e., the sequence (18) is exact.

• (2) The sequence (17) is exact.

• (3) The submodule $M = {{\,\textrm{Im}\,}}_R(A^T)$ is torsion-free.

• (4) The submodule $M = {{\,\textrm{Im}\,}}_R(A^T)$ is $\{0\}$-primary.

Proof

The equivalence of (1) and (2) holds because ${\mathcal {F}}$ is an injective cogenerator. The equivalence of (2), (3), (4) is a standard result in commutative algebra, also found in [10]. $\square$

If the conditions in Theorem 3.3 are met then we have a parametrization of all solutions:

$$\begin{aligned} \,{{\,\textrm{Sol}\,}}_{\mathcal {F}}(A) \,\,=\,\, {{\,\textrm{Im}\,}}_{{\mathcal {F}}}(B) \,=\, \bigl \{B \bullet \psi :\psi \in {\mathcal {F}}^{k'}\bigr \}. \end{aligned}$$

(19)

In general, ${{\,\textrm{Sol}\,}}_{\mathcal {F}}(A)$ is strictly contained in ${{\,\textrm{Im}\,}}_{{\mathcal {F}}}(B)$: not all solutions of A are in the image of B. In that case, the operator can be split into two operators $A_0$ and $A_1$, where $A_0$ admits a vector potential and $A_1$ does not, in the sense that for all $B \in R^{k \times k'}$, there exists $\psi \in {\mathcal {F}}^{k'}$ such that $B \bullet \psi \not \in {{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(A_1)$. This condition is equivalent to $\{0\} \not \in \mathop {\textrm{Ass}}\limits (M_1)$ for $M_1 = {{\,\textrm{Im}\,}}_R(A_1^T)$. It is also equivalent to $R^k/M_1$ being a torsion module.

We write $M = M_0 \cap M_1$, where $M_0$ is $\{0\}$-primary, and $\{0\} \not \in \mathop {\textrm{Ass}}\limits (M_1)$. This is obtained from a primary decomposition of M, where $M_1$ is the intersection of all primary components that are not $\{0\}$-primary. The solutions can be decomposed as ${{\,\textrm{Sol}\,}}_{\mathcal {F}}(M) = {{\,\textrm{Sol}\,}}_{\mathcal {F}}(M_0) + {{\,\textrm{Sol}\,}}_{\mathcal {F}}(M_1)$, which is known in control theory [10] as the controllable–uncontrollable decomposition.

Theorem 3.4

The PDE A has compactly supported solutions if and only if $\{ 0 \} {\in } \mathop {\textrm{Ass}}\limits (M)$.

Proof

This result is contained in [10, §3]. We offer a short proof. Suppose $\{0\} \in \mathop {\textrm{Ass}}\limits (M)$. Then M is a submodule of a $\{0\}$-primary module $M_0 \subseteq R^k$. Write $M_0$ as the R-row span of a matrix $A_0$ and let $B_1$ be any column in its syzygy matrix B. Then for any compactly supported distribution $\psi$, the solution $B_1 \bullet \psi \in {{\,\textrm{Sol}\,}}_{{\mathcal {F}}}(A)$ is also compactly supported.

For the converse, suppose $\{0\} \not \in \mathop {\textrm{Ass}}\limits (M)$. Let $\phi \in {\mathcal {F}}^k \backslash \{0\}$ be a compactly supported solution. By Proposition 3.1, there exists $f \in R^k \backslash M$ such that $f^T \bullet \phi \ne 0$. Since $R^k/M$ is torsion, $rf \in M$ for some nonzero $r \in R$. Thus $r \bullet f^T \bullet \phi = 0$. Taking Fourier transforms, by the Paley–Wiener–Schwartz Theorem [5, Thm. 7.3.1], we get the equation $r(\xi ) \cdot f(\xi )^T {\hat{\phi }}(\xi ) = 0$ of analytic functions. Since $r(\xi ) \ne 0$, we must have $f(\xi )^T {\hat{\phi }}(\xi ) = 0$, a contradiction. $\square$

In conclusion, for any linear PDE A as above, the solution space ${{\,\textrm{Sol}\,}}_{\mathcal {F}}(A)$ decomposes into a subspace ${{\,\textrm{Sol}\,}}_{\mathcal {F}}(A_0) = {{\,\textrm{Im}\,}}_{\mathcal {F}}(B)$ which contains all compactly supported solutions, and another subspace ${{\,\textrm{Sol}\,}}_{\mathcal {F}}(A_1)$, with no compactly supported solutions at all. We are interested in the former solutions, so our computational task is to go from A to B and then to $A_1$. This amounts to two syzygy computations, and is easily carried out in Macaulay2. By applying B to potentials $\psi (z) = \delta (L_1(z),\ldots ,L_{n-r}(z))\cdot e_i$, we obtain interesting solutions to A, as in (10).

4 Waves and Varieties

Our $\ell \times k$ matrix A specifies PDE constraints of order d for distributions $\phi :\mathbb {R}^n \rightarrow \mathbb {C} ^k$. The solution space (15) depends only on the R-module M generated by the rows of A. In this section we introduce several algebraic varieties that are naturally associated with A. As is customary in algebraic geometry, we work in complex projective spaces rather than in real affine spaces. Every subvariety of $\mathbb {P}^{k-1}$ corresponds to a cone in $\mathbb {C} ^k$, which is a complex variety defined by homogeneous equations, and by restricting to $\mathbb {R}^k$ one obtains a real cone. Among such cones are the wave cones from [3] which motivated our study. We shall return to the analytic perspective in the next section. In what follows, however, we stick to algebra. This means working in the projective spaces $\mathbb {P}^{k-1}$ and $\mathbb {P}^{n-1}$ over the complex numbers $\mathbb {C}$.

For any point $y \in \mathbb {P}^{n-1}$ we write A(y) for the complex $\ell \times k$ matrix that is obtained from A by replacing each $\partial _i$ with the coordinate $y_i$. The matrix A(y) is well-defined up to scale. We view it as a point in the projective space $\mathbb {P}^{\ell k - 1}$. We write z for points in $\mathbb {P}^{k-1}$, and we set

$$\begin{aligned} {\mathcal {I}}_A \,\,\,= \,\,\, \bigl \{\,(y,z) \in \mathbb {P}^{n-1} \times \mathbb {P}^{k-1}:\, A(y) \cdot z = 0 \,\bigr \}. \end{aligned}$$

(20)

This is our algebro-geometric representation of the relation between frequencies and amplitudes seen in (4). The projection of the incidence variety ${\mathcal {I}}_A$ onto the first factor equals

$$\begin{aligned} {\mathcal {S}}_A \,\,\, = \,\,\, \{ \, y \in \mathbb {P}^{n-1} : \, \textrm{rank}(A(y)) \le k-1 \,\}. \end{aligned}$$

(21)

This projective variety is the support of our PDE A, and only depends on the module M. The notation V(M) was used in [1, Sect. 3] for the affine cone over ${\mathcal {S}}_A$. The role of the support for simple waves was highlighted in [1, Lemma 3.6]. A natural set of polynomials that define ${\mathcal {S}}_A$ set-theoretically is the $k \times k$ minors of A. However, these minors usually do not suffice to generate the radical ideal of ${\mathcal {S}}_A$. There are two interesting extreme cases, namely ${\mathcal {S}}_A = \mathbb {P}^{n-1}$ and ${\mathcal {S}}_A = \emptyset$. The former identifies PDEs with compactly supported solutions (cf. Theorem 3.4), while the latter identifies PDEs whose only solutions are polynomials [1, Theorem 3.8].

We next consider the projection of the incidence variety ${\mathcal {I}}_A$ onto the second factor $\mathbb {P}^{k-1}$. The resulting projective variety is called the wave variety of A, and we write it as follows:

$$\begin{aligned} {\mathcal {W}}_A \,\,\,:= \,\, \bigcup _{y \in \mathbb {P}^{n-1} } \ker A(y). \end{aligned}$$

The kernel in this definition is a linear subspace of $\mathbb {P}^{k-1}$, so ${\mathcal {W}}_A$ is a projective variety in $\mathbb {P}^{k-1}$. This is the algebraic variant of the wave cone considered in analysis; see [4, Theorem 1.1] and surrounding references. We shall return to this in Sect. 5, where it is denoted ${\mathcal {W}}_{A,\mathbb {R}}$.

Example 4.1

($n= k=\ell =3,d=2$) Consider the second order PDE given by the matrix

$$\begin{aligned} A \,\,\, = \,\,\, \begin{bmatrix} \partial _1^2 &{} \partial _2^2 &{}\,\, \partial _3^2 \,\,\,\\ -\partial _2^2 &{} \partial _3^2 &{} \,\,\partial _1^2\, \,\,\\ -\partial _3^2 &{} -\partial _1^2 &{}\,\, \partial _2^2\,\,\, \end{bmatrix}\!. \end{aligned}$$

Its support ${\mathcal {S}}_A$ is the smooth sextic curve in $\mathbb {P}^2$ defined by $\textrm{det}(A(y)) = y_1^6+y_2^6+y_3^6+y_1^2 y_2^2 y_3^2$. The wave variety ${\mathcal {W}}_A$ is the smooth cubic curve in $\mathbb {P}^2$ defined by $z_1^3 - z_2^3 + z_3^3 - z_1z_2z_3$. These two plane curves are linked by their incidence curve ${\mathcal {I}}_A \subset \mathbb {P}^2 \times \mathbb {P}^2$. If the entries of A are replaced by random quadrics in $\partial _1,\partial _2,\partial _3$, then ${\mathcal {W}}_A$ is a singular curve of degree 12 in $\mathbb {P}^2$. $\triangle$

The article [3] extended the results in [4] by introducing two refined notions of wave cones. We now recast these as algebraic varieties. For $r \in \{0,\ldots ,n-1\}$, the level r wave variety is

$$\begin{aligned} {\mathcal {W}}^r_A \,\,\,:= \bigcup _{\pi \in {\text {Gr}}(n-r, n)} \bigcap _{ y \in \pi } \, \ker A(y). \end{aligned}$$

(22)

The union is over the Grassmannian ${\text {Gr}}(n-r, n)$ of linear subspaces $\pi$ of codimension r in $\mathbb {P}^{n-1}$. For basics on Grassmannians and their projective embeddings see [15, Chapter 5]. For $r=n-1$, the inner intersection in (22) goes away, the outer union is over $y \in \mathbb {P}^{n-1}$, and we obtain the wave variety ${\mathcal {W}}_A$. At the other end of the spectrum, the level 0 wave variety ${\mathcal {W}}^0_A \,\,\, = \bigcap _{y \in \mathbb {P}^{n-1} } \ker A(y)$ is often empty. For the in-between levels r, we obtain a hierarchy

$$\begin{aligned} {\mathcal {W}}^0_A \,\subseteq \, {\mathcal {W}}^1_A \, \subseteq \cdots \,\subseteq \,{\mathcal {W}}^{n-1}_A \, =\, {\mathcal {W}}_A \,\,\subseteq \, \, \mathbb {P}^{k-1}. \end{aligned}$$

(23)

We now define a second hierarchy in $\mathbb {P}^{k-1}$ by switching the intersections and the union. Namely, for any integer $r \in \{1,\ldots ,n\}$, we define the level r obstruction variety to be

$$\begin{aligned} {\mathcal {O}}^r_A \,\,\,:= \bigcap _{\sigma \in {\text {Gr}}(r, n)} \bigcup _{ y \in \sigma }\, \ker A(y). \end{aligned}$$

(24)

This intersection is over the Grassmannian of $(r-1)$-dimensional subspaces in $\mathbb {P}^{n-1}$. The smallest and the largest obstruction variety coincides with the corresponding wave variety.

Lemma 4.2

We have the inclusions ${\mathcal {W}}^r_A \subseteq {\mathcal {O}}^{r+1}_A$ for all r, with ${\mathcal {W}}^0_A = {\mathcal {O}}^1_A$ and ${\mathcal {W}}^{n-1}_A = {\mathcal {O}}^n_A$.

Proof

Fix $z \in {\mathcal {W}}^r_A$ and a codimension r subspace $\pi$ of $\mathbb {P}^{n-1}$ such that $A(y) z = 0$ for all $y \in \pi$. Consider any r-dimensional subspace $\sigma$ of $\mathbb {P}^{n-1}$. Pick a point w in the intersection $\pi \cap \sigma$. Since $A(w) z = 0$, we have $z \in \bigcup _{y \in \sigma } \textrm{ker} A(y)$, and hence $z \in {\mathcal {O}}_A^{r+1}$. Equality holds for $r = 0$ because ${\mathcal {W}}^0_A = \bigcap _{y \in \mathbb {P}^{n-1}} \textrm{ker} A(y) = {\mathcal {O}}^1_A$, and for $r = n-1$ because ${\mathcal {W}}^{n-1}_A = \bigcup _{y \in \mathbb {P}^{n-1}} \textrm{ker} A(y) = {\mathcal {O}}^n_A$. $\square$

In analogy to the wave varieties in (23), there is also a hierarchy of obstruction varieties:

$$\begin{aligned} {\mathcal {W}}^0_A \,= \, {\mathcal {O}}^1_A \,\subseteq \, {\mathcal {O}}^2_A \,\subseteq \cdots \,\subseteq \,{\mathcal {O}}^n_A \, =\, {\mathcal {W}}_A \,\,\subseteq \, \, \mathbb {P}^{k-1}. \end{aligned}$$

(25)

Example 4.3

($n=3, k=4, r=2$) Fix the matrix A in Example 1.1 and 2.5. For every $z \in \mathbb {P}^3$, there exists $y \in \mathbb {P}^2$ with $A(y) z = 0$, and hence ${\mathcal {W}}^2_A = {\mathcal {O}}^3_A = \mathbb {P}^3$. But, for every z, there also exists $y \in \mathbb {P}^2$ with $A(y) z \not =0$, and hence ${\mathcal {W}}^0_A = {\mathcal {O}}^1_A = \emptyset$. The variety in the middle of (23) and (25) satisfies ${\mathcal {W}}^1_A = {\mathcal {O}}^2_A \subset \mathbb {P}^3$. This is the twisted cubic curve $z = (s^3,s^2t,st^2,t^3)$. Indeed, the matrix $\left( {\begin{array}{c}\,z_1 \,\, z_2 \,\, z_3 \,\\ \, z_2 \,\,z_3 \,\, z_4 \,\end{array}}\right)$ has rank 1, with kernel $\pi = \{ y \in \mathbb {P}^2 \,: \, s^2 y_1 + st y_2 + t^2 y_3 = 0 \} \in \textrm{Gr}(2,3)$. Every other line $\sigma \in \textrm{Gr}(2,3)$ in the projective plane $\mathbb {P}^2$ intersects the line $\pi$. $\triangle$

We next recall a basic construction from algebraic geometry; see [12, Example 6.19]. Fix a projective variety $X \subset \mathbb {P}^{n-1}$. The Fano variety $\textrm{Fano}_r(X)$ is the subvariety of the Grassmannian $\textrm{Gr}(n-r,n)$ whose points are the linear spaces $\pi$ of codimension r in $\mathbb {P}^{n-1}$ that lie in X. We use Fano varieties to argue that the inclusion in Lemma 4.2 can be strict.

Example 4.4

($k=\ell =1, n \ge 3$) A subvariety of $\mathbb {P}^0$ is either empty or a point. Let $A = [a]$, where a is irreducible of degree d. Then $\textrm{Fano}_1(X) = \emptyset$. Our varieties in (22) and (24) are

$$\begin{aligned} {\mathcal {W}}^r_A \, = \, {\left\{ \begin{array}{ll} \,\emptyset &{} \textrm{if}\,\, \textrm{Fano}_r(X) = \emptyset , \\ \, \mathbb {P}^0 &{} \textrm{if}\,\, \textrm{Fano}_r(X) \not = \emptyset , \end{array}\right. } \qquad \textrm{and} \qquad {\mathcal {O}}^{r+1}_A \, = \, {\left\{ \begin{array}{ll} \,\emptyset &{} \textrm{if}\,\, r=0, \\ \, \mathbb {P}^0 &{} \textrm{if}\,\, r \ge 1. \end{array}\right. } \end{aligned}$$

If $d \ge 2$ then $\textrm{Fano}_1(X) = \emptyset$, so ${\mathcal {W}}^1_A$ is strictly contained in ${\mathcal {O}}^2_A$. Equality holds for $d=1$. $\triangle$

Returning to arbitrary k and $\ell$, we now show that equality always holds for first-order PDEs. The main point for $d=1$ is this: we can write the product $\,A(y) z \,$ as $\, C(z) y \,$, where C(z) is an $\ell \times n$-matrix whose entries are linear forms in $z_1,\ldots ,z_k$. We did this in (12).

Proposition 4.5

If $d=1$ then ${\mathcal {W}}^r_A = {\mathcal {O}}^{r+1}_A = \bigl \{ z \in \mathbb {P}^{k-1} \,: \, \textrm{rank}(C(z)) \le r \bigr \}\,$ for all r.

Proof

Fix $z \in \mathbb {P}^{k-1}$. The condition $z \in {\mathcal {W}}^r_A$ says that the kernel of the matrix C(z) contains a subspace $\pi$ of codimension r. The condition $z \in {\mathcal {O}}^{r+1}_A$ says that the kernel of C(z) meets every r-dimensional subspace $\sigma$ of $\mathbb {P}^{n-1}$. Both conditions are equivalent to $\textrm{rank}(C(z)) \le r$. $\square$

Thus, the wave varieties of first-order PDEs are easy to write down: they are the determinantal varieties of the auxiliary matrix C(z). For $d \ge 2$, elimination methods from nonlinear algebra (e.g., Gröbner bases) are needed to compute the defining equations of these varieties.

Proposition 4.6

The wave varieties ${\mathcal {W}}^r_A$ and the obstruction variety ${\mathcal {O}}^r_A$ are indeed varieties in the projective space $\mathbb {P}^{k-1}$, i.e., they are zero sets of homogeneous polynomials in k variables.

Proof

The following incidence variety is closed in its ambient product space:

$$\begin{aligned} {\mathcal {I}}^r_A \,\, = \,\, \bigl \{ \,(y,z,\pi ) \in \mathbb {P}^{n-1} \times \mathbb {P}^{k-1} \times \textrm{Gr}(n-r,n) :\, A(y) z = 0 \,\,\, \textrm{and} \, \,\,y \in \pi \,\bigr \}.\nonumber \\ \end{aligned}$$

(26)

The sets we defined in (22) and (24) are derived from this variety by quantifier elimination:

$$\begin{aligned} {\mathcal {W}}^r_A \, = \, \bigl \{ \,z : \,\exists \pi \, \,\forall \,y \,\, (y,z,\pi ) \in {\mathcal {I}}^r_A \,\bigr \} \quad \textrm{and} \quad {\mathcal {O}}^r_A \, = \, \bigl \{ \,z : \,\forall \pi \, \,\exists \,y \, \,(y,z,\pi ) \in {\mathcal {I}}^r_A \,\bigr \}. \end{aligned}$$

These two sets are closed in $\mathbb {P}^{k-1}$ because all their defining equations are homogeneous in each group of variables. For the existential quantifier this follows from the Main Theorem of Elimination Theory [15, Theorem 4.22]. For the universal quantifier once checks it directly.

We compute ideals for ${\mathcal {W}}^r_A$ and ${\mathcal {O}}^r_A$ as follows. The equations $A(y) z= 0$ are bihomogeneous of degree (d, 1). The condition $y \in \pi$ translates into bilinear equations in (y, p), where p is the vector of Plücker coordinates of $\pi$. We view these as equations in y with coefficients in (z, p), and we form the ideal of all coefficient polynomials. The zero set of this ideal is the subvariety $\,\bigcap _{ y \in \pi } \textrm{ker} A(y)$, which lies in $\textrm{Gr}(r,n) \times \mathbb {P}^{k-1}$. We now project that variety onto the second factor to obtain ${\mathcal {W}}^r_A$. This amounts to saturating and then eliminating the Plücker coordinates p. What arises is an ideal in the unknowns z whose zero set is ${\mathcal {W}}^r_A$.

To get the ideal of ${\mathcal {O}}^r_A$, we modify the argument as follows. Again, we consider a fixed but unknown Plücker vector p and we consider the equations for $y \in \pi$ along with $A(y) z = 0$. From these equations we eliminate y to obtain polynomials in (p, z) whose zero set is $\bigcup _{y \in \pi } \ker A(y)$. We now vary p and we view this as a subvariety of $\textrm{Gr}(r,n) \times \mathbb {P}^{k-1}$. We consider the defining equations of this subvariety, and we write them as polynomials in p whose coefficients are polynomials in z. The collection of all such coefficient polynomials defines a subvariety of $\mathbb {P}^{k-1}$. By construction, that subvariety equals the desired set ${\mathcal {O}}^r_A$. $\square$

5 Back to Analysis

We now return to the setting of waves $\phi : \mathbb {R}^n \rightarrow \mathbb {C} ^k$ that was introduced in Sect. 2. The projective varieties ${\mathcal {W}}^r_A$ and ${\mathcal {O}}^r_A$ in $\mathbb {P}^{k-1}$ are to be viewed as affine cones in $\mathbb {C} ^{k}$. We write

$$\begin{aligned} {\mathcal {W}}_{A,\mathbb {R}}^r \,\,\,&{:}{=}\bigcup _{\pi \in {{\,\textrm{Gr}\,}}_\mathbb {R}(n-r,n)} \bigcap _{y \in \pi \setminus \{0\}} \ker A(y), \\ {\mathcal {O}}_{A,\mathbb {R}}^r \,\,\,&{:}{=}\,\, \,\bigcap _{\sigma \in {{\,\textrm{Gr}\,}}_\mathbb {R}(r,n)} \, \,\bigcup _{y \in \sigma \setminus \{0\}} \ker A(y), \end{aligned}$$

where ${{\,\textrm{Gr}\,}}_\mathbb {R}(r,n)$ is the Grassmannian of r-dimensional subspaces in $\mathbb {R}^n$. In these definitions, the kernel of A(y) is over the complex numbers, but $\pi$ and $\sigma$ are required to be real. Hence ${\mathcal {W}}_{A,\mathbb {R}}^r$ and ${\mathcal {O}}_{A,\mathbb {R}}^r$ are subsets in $\mathbb {C} ^k$, closely related to the projective varieties in (22) and (24).

Readers of [3] will note that we changed notation and nomenclature. The $\ell$-wave cone $\Lambda ^\ell _{\mathcal {A}}$ from [3, Definition 1.2] is the obstruction cone ${\mathcal {O}}^r_{A,\mathbb {R}}$ here, while the cone ${\mathcal {N}}^\ell _{\mathcal {A}}$ defined later in [3, eqn (1.8)] is our wave cone ${\mathcal {W}}^r_{A,\mathbb {R}}$. The coming results will motivate these choices.

Proposition 2.2 shows why ${\mathcal {W}}^r_{A,\mathbb {R}}$ serves as the rth wave cone. The distribution in (10) has the form $\, \mathbb {R}^n \rightarrow \mathbb {C} ^k \,: x \,\mapsto \, \delta (L x) \cdot u$, where L is the $(n-r) \times n$ matrix whose rows are the coefficients of $L_1,\ldots ,L_{n-r}$. Recall Remark 2.3 for the definition of $\delta (Lx)$ as a distribution.

Proposition 5.1

A vector $u\in \mathbb {C} ^k$ lies in the wave cone ${\mathcal {W}}^r_{A,\mathbb {R}}$ if and only if there is a matrix $L \in \mathbb {R}^{(n-r) \times n}$ such that $x \mapsto \delta (L x) \cdot u$ is a solution to A for all distributions $\delta \in {\mathcal {D}}'( \mathbb {R}^{n-r}, \mathbb {C} )$.

Proof

By definition, a complex vector u lies in the wave cone ${\mathcal {W}}^r_{A,\mathbb {R}}$ if and only if there exists a real subspace $\pi \in \textrm{Gr}_\mathbb {R}(n-r,n)$ such that $A(\xi ) u = 0$ for all $\xi \in \pi \subseteq \mathbb {R}^n$. This is equivalent to saying that $(u, \pi )$ is a wave pair for A. If we identify $\pi$ with the rowspace of L, then the result follows from Proposition 2.2. $\square$

We next present an analogous statement for the obstruction cones ${\mathcal {O}}^r_{A,\mathbb {R}}$.

Proposition 5.2

A vector $u \in \mathbb {C} ^k$ lies in ${\mathcal {O}}^r_{A,\mathbb {R}}$ if and only if, for all $S \in \mathbb {R}^{r \times n}$ of rank r, the PDE A has a wave solution $\,x \mapsto \delta ( Sx) \cdot u$, where $\delta$ is nonconstant and bounded.

Proof

Suppose $u \in {\mathcal {O}}^r_{A,\mathbb {R}}$ and let $\sigma \in \textrm{Gr}_\mathbb {R}(r,n)$ be the real rowspan of the real matrix S. Fix a nonzero vector $\xi \in \sigma$ such that $A(\xi )u = 0$, and let $\eta \in \mathbb {R}^r$ such that $\xi = \eta S$. The exponential function $\delta _\eta (t) = \textrm{exp}(i \eta \cdot t)$ is nonconstant and bounded. Moreover, the function $\delta _\eta (Sx) \cdot u$ is a wave solution to the PDE A, by the same calculation as in the proof of Proposition 5.1. This proves the only-if direction.

For the if direction, let $u \notin {\mathcal {O}}^r_{A,\mathbb {R}}$. There exists $\sigma \in \textrm{Gr}_{\mathbb {R}}(r,n)$ such that $A(\xi )\cdot u\ne 0$ for all $\xi \in \sigma \backslash \{0\}$. Let S be as before the real matrix with rowspan $\sigma$. Now suppose $\delta (Sx)\cdot u$ is a bounded solution of A. By the proof of Proposition 2.2, this implies that $\delta (y)$ is a bounded solution of the operator $\alpha (\partial _y)=A(\partial _y S) \cdot u$. This operator is elliptic by our assumption. By classical theory (cf. [11, Theorem 2-7]), every solution to $\alpha \bullet v=f$ with $f\in C^\infty$ is in $C^\infty$. Therefore a Liouville theorem holds: one can use the Closed Graph Theorem to deduce that there is a constant $C>0$ such that for any solution of $\alpha \bullet v =0$ in the unit ball $B_1$ one has

$$\begin{aligned} \Vert Dv\Vert _{L^\infty (B_{1/2})} \,\le \, C \Vert v\Vert _{L^\infty (B_1) }. \end{aligned}$$

Since the operator $\alpha$ is of homogenous degree d, we can use scaling to obtain

$$\begin{aligned} \Vert Dv\Vert _{L^\infty (B_R)}\, \,\le \, \frac{C}{R}\Vert v\Vert _{L^\infty (B_{2R})}. \end{aligned}$$

Hence, every bounded solution on $\mathbb {R}^{n-r}$ is constant (cf. [11, Chapter 2]). So, $\delta$ is constant. $\square$

We used the term “obstruction” for the variety ${\mathcal {O}}^r_A$ and the cone ${\mathcal {O}}^r_{A,\mathbb {R}}$ not because their elements are obstructions. Rather, our choice of name refers to role played by the cone ${\mathcal {O}}^r_{A,\mathbb {R}}$ in the paper [3], which motivated us. Since ${\mathcal {O}}^{r}_{A,\mathbb {R}}$ contains the wave cone ${\mathcal {W}}^{r-1}_{A,\mathbb {R}}$, the latter is empty if the former is empty. Thus, the cone ${\mathcal {O}}^{r}_{A,\mathbb {R}}$ being empty is an obstruction to the existence of wave solutions. That obstruction is a key for the “dimensional estimates” in [3].

In the present paper we often transition between real numbers and complex numbers. This occurs at multiple mathematical levels, including trigonometry and projective geometry. The complex numbers represent waves in Sect. 2 and they serve as an algebraically closed field in Sect. 4. However, the argument x of our solutions $\phi (x)$ are real vectors. The spaces (8) belong to the field real analysis, as does the study of A-free Radon measures in [3, 4, 21]. Recall that a Radon measure is a distribution that admits an integral representation, and one is interested in rectifiability of such measures that satisfy the PDE constraint given by A.

This raises the question of how complex analysis fits in. From a purely algebraic point of view, we can certainly consider solutions in the space of holomorphic functions $\phi : \mathbb {C} ^n \rightarrow \mathbb {C} ^k$. All our formal results extend gracefully to that setting. For instance, we can certainly take $\delta$ in (10) to be a holomorphic function on $\mathbb {C} ^{n-r}$ to get a holomorphic solution $\phi$ to our PDE. However, from an analytic point of view, there are no meaningful waves in complex analysis. The following example is meant to illustrate the importance of reality for making waves.

Example 5.3

($n=2,k=\ell =1, \, d=1,2$) We consider PDEs for scalar-valued functions in two variables. The transport equation $A = \partial _1+\partial _2$ has the solutions $\delta (x_1-x_2)$. These are waves and $\delta$ can be any distribution. The Cauchy–Riemann equation $A' = \partial _1 + i \partial _2$ looks very similar, and we can write its solutions formally as $\delta (x_1 + ix_2)$. But, these solutions do not come from the wave cone ${\mathcal {W}}^r_{A,\mathbb {R}}$, since here $\pi$ is not real, and these do not give waves. Passing to second-order equations, one might compare $\partial _1^2- \partial _2^2$ and $\partial _1^2+\partial _2^2$. These two PDE look indistinguishable to the eyes of an algebraist, while an analyst will see a hyperbolic PDE and an elliptic PDE. These two classes have vastly different properties for their solutions. In particular, the latter can only admit smooth solutions. $\triangle$

The affine cones ${\mathcal {W}}_{A,\mathbb {R}}^r$ and ${\mathcal {O}}_{A,\mathbb {R}}^r$ can be quite different from the complex varieties ${\mathcal {W}}_A^r$ and ${\mathcal {O}}_A^r$. In general we have ${\mathcal {W}}_A^r \supseteq {\mathcal {W}}_{A,\mathbb {R}}^r$. Indeed, if $z \in {\mathcal {W}}_A^r$, there is a linear subspace $\pi \in {{\,\textrm{Gr}\,}}(n-r,n)$ such that $A(y)z = 0$ for all $y \in \pi$. For $z \in \mathbb {C} ^k$ to lie in ${\mathcal {W}}_{A,\mathbb {R}}^r$, we must impose the additional condition that the dimension of $\pi \cap \mathbb {R}^n$ is also $n-r$. The inclusions for the obstruction cones are reversed: ${\mathcal {O}}_A^r \subseteq {\mathcal {O}}_{A,\mathbb {R}}^r$. The point $z \in \mathbb {C} ^k$ lies in ${\mathcal {O}}_A^r$ if and only if for all $\sigma \in {{\,\textrm{Gr}\,}}(r,n)$ there exists $y\in \sigma \setminus \{0\}$ such that $A(y)z = 0$. This condition is relaxed in ${\mathcal {O}}_{A,\mathbb {R}}^r$, where it suffices to consider those $\sigma$ whose real part $\sigma \cap \mathbb {R}^n$ also has dimension r.

We close with an example that highlights the connection to the theory of rank-one convexity in the study of nonlinear PDE [17]. Here, one is interested in solutions to A that additionally satisfy differential inclusions $\phi (x) \in {\mathcal {K}}$, where ${\mathcal {K}}$ is a specified subset of $\mathbb {C} ^k$. Of special interest is the case when the target is a matrix space and ${\mathcal {K}}$ is a finite set of matrices.

Example 5.4

($n=2, d=1, k=3,\ell = 2$) Consider the action of the curl operator on the 3-dimensional space of symmetric $2 \times 2$-matrices $\left( {\begin{array}{c}\,\phi _1 \,\, \phi _2 \,\\ \, \phi _2 \,\,\phi _3 \,\end{array}}\right)$. In our notation, this corresponds to

This PDE is a simplified version of that in Examples 1.1, 2.5, and 4.3. The wave cone consists of symmetric $2 \times 2$ matrices of rank 1. Here, ${\mathcal {K}}$ is a finite set in $\mathbb {R}^3$, such as the five matrices in [18], whose rank-one convex hull is of great interest. Our varieties in Sect. 4 offer an algebraic framework for higher notions of convexity that might be of interest in analysis. $\triangle$

6 Computing Wave Pairs

Our aim is to solve a PDE, given by an $\ell \times k$ matrix A whose entries are homogeneous polynomials of degree d in $R = \mathbb {C} [\partial _1,\ldots ,\partial _n]$. Each wave (10) arises from a wave pair $(z,\pi )$, which serves as a blueprint for creating solutions to the PDE. Our approach allows complete freedom in making waves with desirable analytic properties, by choosing the distribution $\delta$ in Proposition 5.1. Inspired by Proposition 2.2, we define the wave pair variety

$$\begin{aligned} {\mathcal {P}}^r_A \,\, = \,\, \bigl \{ (z, \pi ) \in \mathbb {P}^{k-1} \times {{\,\textrm{Gr}\,}}(n-r, n) \,:A(y)z = 0 \,\,\, \hbox {for all}\, \,\,y \in \pi \bigr \}. \end{aligned}$$

This is a smaller version of the incidence variety ${\mathcal {I}}^r_A$ we saw in (26). The wave variety ${\mathcal {W}}^r_A$ introduced in (22) is the projection of the wave pair variety ${\mathcal {P}}^r_A$ onto the first factor $\mathbb {P}^{k-1}$. For $r=n-1$ the wave pair variety coindices with the incidence variety in (20), that is

$$\begin{aligned} {\mathcal {P}}^{n-1}_A \,\,=\,\, {\mathcal {I}}_A. \end{aligned}$$

(27)

It is instructive to start with the case $k=1$. Here, ${\mathcal {P}}^r_A$ lives in $\mathbb {P}^0 \times {{\,\textrm{Gr}\,}}(n-r,n)$, which we identify with ${{\,\textrm{Gr}\,}}(n-r,n)$. Consider the subvariety ${\mathcal {S}}_A$ of $\mathbb {P}^{n-1}$ that is defined by the $\ell$ entries of the $\ell \times 1$ matrix A. This is the support of our PDE, as seen in (21). The condition $A(y) z = 0$ for $z \in \mathbb {P}^0$ simply means that $y \in {\mathcal {S}}_A$. From this we conclude the following fact.

Corollary 6.1

If $k=1$ then ${\mathcal {P}}^r_A = \textrm{Fano}_{r}({\mathcal {S}}_A)$ is the Fano variety of the support ${\mathcal {S}}_A$. The points of ${\mathcal {P}}^r_A$ are the linear spaces of codimension r in $\mathbb {P}^{n-1}$ that are contained in ${\mathcal {S}}_A$.

The software Macaulay2 has a built-in command Fano for computing the ideal of the Fano variety $\textrm{Fano}_{r}({\mathcal {S}}_A)$ from the entries of A. Our results in this section extend this method. We shall describe an algorithm for computing ${\mathcal {P}}^r_A$ and all the varieties introduced in Sect. 4.

Each of our varieties lies in a projective space or product of projective spaces. What we seek is its saturated ideal. To explain what this means, consider the variety ${\mathcal {I}}_A$ in $\mathbb {P}^{n-1} \times \mathbb {P}^{k-1}$ described in (20). The $\ell$ coordinates of A(y)z are polynomials of bidegree (d, 1) in

$$\begin{aligned} \mathbb {C} [y,z] \,\, = \,\, \mathbb {C} [y_1,\ldots ,y_n,z_1,\ldots ,z_k]. \end{aligned}$$

However, these $\ell$ polynomials do not suffice. The saturated ideal of the variety ${\mathcal {I}}_A$ equals

$$\begin{aligned} \bigl ( \,\bigl ( \,\langle \,A(y) z \,\rangle : \langle y_1,\ldots ,y_n \rangle ^\infty \, \bigr ): \langle z_1,\ldots ,z_k \rangle ^\infty \,\bigr ). \end{aligned}$$

(28)

Saturation is a built-in command in Macaulay2 [20], but its execution often takes a long time. This crucial step removes extraneous contributions by the irrelevant ideals of $\mathbb {P}^{n-1}$ and $\mathbb {P}^{k-1}$.

Example 6.2

($k=\ell =n=d=2$) The parameters are as in Example 3.2, but now the entries of A are general quadrics in $\mathbb {C} [y_1,y_2]$. The variety ${\mathcal {I}}_A$ consists four points in $\mathbb {P}^1 \times \mathbb {P}^1$. Its ideal (28) is generated by six polynomials of bidegrees (0, 4), (1, 2), (1, 2), (2, 1), (2, 1), (4, 0). The first and last equation are binary quartics that define the projections ${\mathcal {S}}_A$ and ${\mathcal {W}}_A$ into $\mathbb {P}^1$. These data encode the general solution to the PDE A. For a concrete example, consider

$$\begin{aligned} A \,=\, \begin{bmatrix} \partial _1^2 + 4\partial _2^2 &{} 17\partial _1\partial _2 \\ 2\partial _1\partial _2 &{} 4\partial _1^2 + \partial _2^2 \end{bmatrix}. \end{aligned}$$

Here, the general solution $\phi : \mathbb {R}^2 \rightarrow \mathbb {C} ^2$ is given by the following superposition of waves:

$$\begin{aligned} \phi (x_1,x_2)&= \begin{bmatrix} -17 \\ 4 \end{bmatrix}\alpha (2x_1+x_2) + \begin{bmatrix} 17 \\ 4 \end{bmatrix}\beta (-2x_1+x_2) \\&\quad + \begin{bmatrix} -2 \\ 1 \end{bmatrix}\gamma (x_1+2x_2) + \begin{bmatrix} 2 \\ 1 \end{bmatrix}\delta (x_1-2x_2), \end{aligned}$$

where $\alpha ,\beta ,\gamma ,\delta \in {\mathcal {D}}'$. This can also be found using the methods described in [1]. $\triangle$

The points $\pi$ in the Grassmannian ${{\,\textrm{Gr}\,}}(n-r,n)$ will be represented as in [15, Sect. 5.1]. We write $\pi$ as the rowspace of an $(n -r) \times n$ matrix $S = (s_{ij})$, that is, $\pi = \{ wS : \,w \in \mathbb {C} ^{n-r}\}$. For a subset I of cardinality $n-r$ in $\{1,\ldots ,n\}$, the corresponding subdeterminant of S is denoted $p_I$. Then $p = (p_I) \in \mathbb {C} ^{\left( {\begin{array}{c}n\\ r\end{array}}\right) }$ is the vector of Plücker coordinates of $\pi$. The resulting embedding of ${{\,\textrm{Gr}\,}}(n-r,n)$ into $\mathbb {P}^{\left( {\begin{array}{c}n\\ r\end{array}}\right) -1}$ is defined by the ideal G of quadratic Plücker relations [15, Sect. 5.2]. Subvarieties of ${{\,\textrm{Gr}\,}}(n-r,n)$ are represented by saturated ideals in $\mathbb {C} [p]/G$. In the special case $r=n-1$, we identify the Plücker coordinates p with $y=(y_1,\ldots ,y_n)$.

The wave pair variety ${\mathcal {P}}_A^r$ lives in $\mathbb {P}^{k-1} \times \mathbb {P}^{\left( {\begin{array}{c}n\\ r\end{array}}\right) -1}$. We shall compute its saturated ideal in the polynomial ring $\mathbb {C} [z,p]/G$. A pair $(z,\pi )$ lies in ${\mathcal {P}}_A^r$ if and only if $A(wS)z = 0$ for all $w \in \mathbb {C} ^{n-r}$. To express this in Plücker coordinates, we proceed as follows. Write the $\ell$ entries of A(wS)z as linear combinations of the monomials $w^\alpha$, $\alpha \in {\mathbb {N}}^{n-r}$, with coefficients in $\mathbb {C} [z,S]$. Let ${\mathcal {J}}$ be the ideal generated by these coefficients, and consider the ring map $\psi : \mathbb {C} [z,p]/G \rightarrow \mathbb {C} [z,S]/{\mathcal {J}}$, which fixes each $z_i$ and maps $p_I$ to the corresponding minor of S.

Algorithm 1

The ideal of the wave pair variety in Plücker coordinates.

To compute the ideal of the wave variety ${\mathcal {W}}_A^r$, one can now eliminate the Plücker variables from the output of Algorithm 1. This corresponds to projecting onto the first factor of ${\mathcal {P}}_A^r$.

We implemented Algorithm 1 in Macaulay2. The code and its documentation are available at the URL

https://mathrepo.mis.mpg.de/makingWaves.

Our command wavePairs(A,r) returns generators of the saturated ideal of ${\mathcal {P}}_A^r$ in $\mathbb {Q}[z,p]/G$, where G is the Plücker ideal, given by the built-in command Grassmannian(n-r-1, n-1). Since Algorithm 1 generalizes the computation of Fano varieties, its execution can take a long time as it requires extensive Gröbner basis computations. A common method to improve executions times is to restrict to an affine patch of the Grassmannian, which can be activated in our implementation by specifying the optional argument Patch. If $\texttt {Patch}$ is set to true, then the leftmost $(n-r) \times (n-r)$ submatrix of S is set to be the identity matrix, as in [15, eqn (5.2)]. Other charts can also be chosen by specifying a list of indices indicating the submatrix of S to set as the identity matrix.

We now come to the special case of first-order PDE ($d=1$). These are ubiquitous in applications, and computing the corresponding wave pair varieties can be simplified due to Proposition 4.5: if C(z) is the $\ell \times n$-matrix given by $\,A(y) z = C(z) y$, then the ${\mathcal {W}}^r_A$ are the determinantal varieties of C(z).

Corollary 6.3

Let $d=1$, with notation as in Proposition 4.5. The wave pair variety equals

$$\begin{aligned} {\mathcal {P}}^r_A \,\, = \,\, \bigl \{ \,(z, \pi ) \in \mathbb {P}^{k-1} \times {{\,\textrm{Gr}\,}}(n-r, n) \,\,:\, \pi \subseteq \textrm{kernel}(C(z)) \,\bigr \}. \end{aligned}$$

If $\pi$ is given as the row space of an $(n-r) \times n$ matrix S then $\pi \subseteq \textrm{kernel}(C(z))$ means that $C(z) \cdot S^T$ is the zero matrix of format $\ell \times (n-r)$. Thus, ${\mathcal {P}}^r_A$ is a vector bundle over the wave variety ${\mathcal {W}}^r_A$. We shall explore these determinantal varieties for some scenarios of geometric origin. These specify PDE which admit interesting wave solutions $x \mapsto \delta (Lx) \cdot u$.

Example 6.4

(Cubic Surfaces) Every smooth cubic surface in $\mathbb {P}^3$ is the determinant of a $3 \times 3$ matrix of linear forms. The surface contains 27 lines, but that number can drop for special cubics. We here present an example with nine lines, namely Cayley’s cubic surface:

$$\begin{aligned} \qquad \qquad A \,\,= \,\, \begin{bmatrix} \partial _1 &{} \partial _2 &{} \partial _3 \\ \partial _2 &{} \partial _1 &{} \partial _4 \\ \partial _3 &{} \partial _4 &{} \partial _1 \end{bmatrix} \,\,,\qquad C \,\,= \,\, \begin{bmatrix} z_1 &{} z_2 &{} z_3 &{} 0 \\ z_2 &{} z_1 &{} 0 &{} z_3 \\ z_3 &{} 0 &{} z_1 &{} z_2 \end{bmatrix} \qquad \qquad (n=4, k=\ell =3). \end{aligned}$$

The only nontrivial wave variety consists of the six points in $\mathbb {P}^2$, where C(z) has rank 2:

$$\begin{aligned} {\mathcal {W}}^2_A \,= \,{\mathcal {O}}^3_A \,=\, \bigl \{ \, (1:1:0), (1:-1:0), (1:0:1), (1: 0: -1), (0: 1: 1), (0: 1: -1) \,\bigr \}.\nonumber \\ \end{aligned}$$

(29)

The cubic surface ${\mathcal {S}}_A= \{\, y\in \mathbb {P}^3: \textrm{det}(A(y)) = 0\, \}$ has four singular points. It is shown in [15, Fig. 1.1]. Geometrically, ${\mathcal {S}}_A$ is the blow-up of $\mathbb {P}^2$ at the six points (29). This is a general fact: if A is a $3 {\times } 3$ matrix representing a cubic surface then its wave solutions are supported on the six lines, among 27, whose blow-down maps the surface birationally onto $\mathbb {P}^2$.

The wave pair variety ${\mathcal {P}}^2_A$ lives in $\mathbb {P}^2 \times \mathbb {P}^5$. Its ideal is the output computed by Algorithm 1:

$$\begin{aligned} \begin{matrix} \langle z_1,z_2{-}z_3,p_{14},p_{23},p_{24}{+}p_{34},p_{13}{-}p_{34},p_{12}{+}p_{34} \rangle \, \cap \, \langle z_1,z_2{+}z_3,p_{14},p_{23},p_{24}{-}p_{34},p_{13}{+}p_{34},p_{12}{+}p_{34} \rangle \, \cap \, \\ \langle z_2,z_1{-}z_3,p_{13},p_{24},p_{14}{+}p_{34},p_{23}{-}p_{34},p_{12}{-}p_{34} \rangle \, \cap \, \langle z_2,z_1{+}z_3,p_{13},p_{24},p_{14}{-}p_{34},p_{23}{+}p_{34},p_{12}{-}p_{34} \rangle \, \cap \, \\ \langle z_3,z_1{-}z_2,p_{12},p_{34},p_{14}{+}p_{24},p_{23}{+}p_{24},p_{13}{-}p_{24} \rangle \, \cap \, \langle z_3,z_1{+}z_2,p_{12},p_{34},p_{14}{-}p_{24},p_{23}{-}p_{24},p_{13}{-}p_{24}\rangle . \end{matrix} \end{aligned}$$

Its projection to $\mathbb {P}^2$ is ${\mathcal {W}}^2_A$, while that to $\mathbb {P}^5$ yields six of the nine points in $\textrm{Fano}_2({\mathcal {S}}_A)$. $\triangle$

We conclude by explicitly computing the wave pair varieties of certain operators that are prominent in the calculus of variations. Such operators are built from $\textsf{div}$, $\textsf{curl}$, and $\textsf{grad}$. We refer to [10, Example 2.1] for a warm-up from the control theory perspective of Sect. 3.

Determinantal varieties are given by imposing rank constraints on matrices [12, Lecture 9]. The following construction realizes such varieties as wave cones of certain natural PDEs.

Example 6.5

(Generic Determinantal Varieties) Let $\textsf{div} = (\partial _1, \partial _2, \ldots ,\partial _n)$, fix $p \ge 2$, and set $k= pn$, $\ell = p$. By taking the p-fold direct sum of $\textsf{div}$, we obtain the first-order PDE

for distributions $\phi : \mathbb {R}^n \rightarrow \mathbb {C} ^{p \times n}$ with coordinates $\phi _{ij}$, where $i=1,\ldots ,p$ and $j=1,\ldots ,n$. The matrix C(z) defined by the bilinear equation $A(y) z = C(z) y$ has format $p \times n$. Its entries are distinct variables $z_{ij}$. The wave variety ${\mathcal {W}}_A^r \subset \mathbb {P}^{pn-1}$ is the determinantal variety of all $p \times n$ matrices z of rank $\le r$. The wave pair variety ${\mathcal {P}}_A^r \subset \mathbb {P}^{pn-1} \times {{\,\textrm{Gr}\,}}(n-r,n)$ consists of pairs $(z,\pi )$, where $\pi$ is in the kernel of z. This is a resolution of singularities for the determinantal variety ${\mathcal {W}}_A^r$. We refer to Examples 12.1 and 16.18 in Harris’ textbook [12]. $\triangle$

We next come to the curl operator, with its action on matrices as in [21, Example 1.16 (c)]. A first glimpse was seen in Example 5.4. Fix any integer $n \ge 2$. We write curl for the $\left( {\begin{array}{c}n\\ 2\end{array}}\right) \times n$ matrix whose rows are vectors $\partial _i e_j - \partial _j e_i$. We take A to be the p-fold direct sum of curl. This matrix has $\ell = p\left( {\begin{array}{c}n\\ 2\end{array}}\right)$ rows and $k = pn$ columns. The following holds for this matrix A.

Proposition 6.6

Let A be the curl operator for distributions $\phi :\mathbb {R}^n \rightarrow \mathbb {C} ^{p\times n}$. The ideal of its wave pair variety ${\mathcal {P}}_A^{n-1} \subseteq \mathbb {P}^{pn-1} \times \mathbb {P}^{n-1}$ is generated by the $2 \times 2$ minors of the $(p{+}1) \times n$ matrix

$$\begin{aligned} \begin{bmatrix} y_1 &{} y_2 &{} \cdots &{} y_n \\ z_{11} &{} z_{12} &{} \cdots &{} z_{1n} \\ z_{21} &{} z_{22} &{} \cdots &{} z_{2n} \\ \vdots &{} \vdots &{} \ddots &{} \vdots \\ z_{p1} &{} z_{p2} &{} \cdots &{} z_{pn} \end{bmatrix}. \end{aligned}$$

(30)

The wave variety ${\mathcal {W}}_A$ is similarly defined by the $2 \times 2$ minors of the $p \times n$ matrix $(z_{ij})$. All other wave pair varieties ${\mathcal {P}}^r_A$ and wave varieties ${\mathcal {W}}^r_A$, indexed by $r \le n-2$, are empty.

Proof

The ideal of the incidence variety ${\mathcal {I}}_A = {\mathcal {P}}_A^{n-1}$ is computed by the saturation (28) from

$$\begin{aligned} \langle \,A(y) z \,\rangle \,\,=\,\, \bigl \langle \,y_i z_{kj} - y_j z_{ki} : \,k = 1,\ldots ,p \,\,\, \textrm{and}\,\,\, 1 \le i < j \le n \,\bigr \rangle . \end{aligned}$$

This step removes contributions from the irrelevant maximal ideal. Every $2 \times 2$ minor of (30) lies in this saturated ideal. Therefore, that ideal equals the prime ideal generated by all the $2 \times 2$ minors of (30). For $r \le n-2$, we note that $A(y)z = C(z)y$ hands us the matrix

$$\begin{aligned} C(z) \,\,\,= \,\,\, - \begin{bmatrix} \textsf{curl}(z_{11},\ldots ,z_{1n}) \\ \textsf{curl}(z_{21},\ldots ,z_{2n}) \\ \cdots \qquad \cdots \\ \textsf{curl}(z_{p1},\ldots ,z_{pn}) \end{bmatrix}. \end{aligned}$$

One checks that this $p \left( {\begin{array}{c}n\\ 2\end{array}}\right) \times n$ matrix cannot have rank $\le n-2$ unless $z_{ij} = 0$ for all i, j. $\square$