# An optimal uncertainty principle in twelve dimensions via modular forms

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## Abstract

We prove an optimal bound in twelve dimensions for the uncertainty principle of Bourgain, Clozel, and Kahane. Suppose \(f :\mathbb {R}^{12} \rightarrow \mathbb {R}\) is an integrable function that is not identically zero. Normalize its Fourier transform \(\widehat{f}\) by \(\widehat{f}(\xi ) = \int _{\mathbb {R}^d} f(x)e^{-2\pi i \langle x, \xi \rangle }\, dx\), and suppose \(\widehat{f}\) is real-valued and integrable. We show that if \(f(0) \le 0\), \(\widehat{f}(0) \le 0\), \(f(x) \ge 0\) for \(|x| \ge r_1\), and \(\widehat{f}(\xi ) \ge 0\) for \(|\xi | \ge r_2\), then \(r_1r_2 \ge 2\), and this bound is sharp. The construction of a function attaining the bound is based on Viazovska’s modular form techniques, and its optimality follows from the existence of the Eisenstein series \(E_6\). No sharp bound is known, or even conjectured, in any other dimension. We also develop a connection with the linear programming bound of Cohn and Elkies, which lets us generalize the sign pattern of *f* and \(\widehat{f}\) to develop a complementary uncertainty principle. This generalization unites the uncertainty principle with the linear programming bound as aspects of a broader theory.

## 1 Introduction

An uncertainty principle expresses a fundamental tradeoff between the properties of a function *f* and its Fourier transform \(\widehat{f}\). The most common variants measure the dispersion, with the tradeoff being that *f* and \(\widehat{f}\) cannot both be highly concentrated near the origin. Motivated by applications to number theory, Bourgain, Clozel, and Kahane [2] proved an elegant uncertainty principle for the signs of *f* and \(\widehat{f}\): if these functions are nonpositive at the origin and not identically zero, then they cannot both be nonnegative outside an arbitrarily small neighborhood of the origin. We can state this principle more formally as follows.

*eventually nonnegative*(resp.,

*nonpositive*) if \(f(x)\ge 0\) (resp., \(f(x)\le 0\)) for all sufficiently large |

*x*|. If that is the case, we let

*f*by

- (1)
\(f\in L^1(\mathbb {R}^d)\), \(\widehat{f}\in L^1(\mathbb {R}^d)\), and \(\widehat{f}\) is real-valued (i.e.,

*f*is even), - (2)
*f*is eventually nonnegative while \(\widehat{f}(0)\le 0\), and - (3)
\(\widehat{f}\) is eventually nonnegative while \(f(0)\le 0\).

*f*and the inequality \(\int _{\mathbb {R}^n} f = \widehat{f}(0) \le 0\), and the analogous tension in (3).)

*r*(

*f*) and \(r(\widehat{f}\,)\) is a natural way to eliminate scale dependence, because rescaling the input of

*f*preserves this quantity. Thus, the uncertainty principle amounts to saying that

*r*(

*f*) and \(r(\widehat{f}\,)\) cannot both be made arbitrarily small if \(f \in \mathcal {A}_+(d){\setminus }\{0\}\).

*d*there exists a radial function \(f\in \mathcal {A}_+(d) {\setminus }\{0\}\) such that \(f = \widehat{f}\) and

*r*(

*g*) in the following optimization problem:

### Problem 1.1

*r*(

*g*) over all \(g :\mathbb {R}^d \rightarrow \mathbb {R}\) such that

- (1)
\(g\in L^1(\mathbb {R}^d){\setminus }\{0\}\) and \(\widehat{g} = g\), and

- (2)
\(g(0)=0\) and

*g*is eventually nonnegative.

The name “\(+\,1\) eigenfunction” refers to the fact that *g* is a eigenfunction of the Fourier transform with eigenvalue \(+\,1\).

Upper and lower bounds for \(\mathrm {A}_+(d)\) are known [2, 12], but the exact value has not previously been determined, or even conjectured, in any dimension. Our main result is a solution of this problem in twelve dimensions:

### Theorem 1.2

*f*has a double root at \(|x|=0\), a single root at \(|x|=\sqrt{2}\), and double roots at \(|x|=\sqrt{2j}\) for integers \(j \ge 2\).

See Fig. 1 for plots. The appealing simplicity of this answer seems to be unique to twelve dimensions, and we have been unable to conjecture a closed form for \(\mathrm {A}_+(d)\) in any other dimension *d*. See Sect. 4 for an account of the numerical evidence, which displays noteworthy patterns and regularity despite the lack of any exact conjectures.

The proof of Theorem 1.2 makes use of modular forms. The lower bound \(\mathrm {A}_+(12) \ge \sqrt{2}\) follows from the existence of the Eisenstein series \(E_6\), while the upper bound \(\mathrm {A}_+(12) \le \sqrt{2}\) is based on Viazovska’s methods, which were developed to solve the sphere packing problem in eight dimensions [18] and twenty-four dimensions [8] (see also [4] for an exposition). We prove both bounds for \(\mathrm {A}_+(12)\) in Sect. 2.

*f*into an upper bound for the sphere packing density \(\Delta _d\) in \(\mathbb {R}^d\). Suppose \(f :\mathbb {R}^d \rightarrow \mathbb {R}\) is an integrable function such that \(\widehat{f}\) is also integrable and real-valued (i.e.,

*f*is even), \(f(0) = \widehat{f}(0) = 1\), \(\widehat{f} \ge 0\) everywhere, and

*f*is eventually nonpositive. Then the linear programming bound obtained from

*f*is the upper bound

*R*about the origin in \(\mathbb {R}^d\). (Strictly speaking, the proof in [5] requires additional decay hypotheses on

*f*and \(\widehat{f}\); see [11, Theorem 3.3] for a proof in the generality of our statement here.) Optimizing this bound amounts to minimizing

*r*(

*f*).

Based on numerical evidence and analogies with other problems in coding theory, Cohn and Elkies conjectured the existence of functions *f* achieving equality in (1.1) when \(d \in \{2,8,24\}\), and they proved it when \(d=1\). The case \(d=2\) remains an open problem today, despite the existence of elementary solutions of the two-dimensional sphere packing problem by other means (see, for example, [13]). However, the case \(d=8\) was proved fourteen years later in a breakthrough by Viazovska [18], and the case \(d=24\) was proved shortly thereafter based on her approach [8]. These papers solved the sphere packing problem in dimensions 8 and 24.

The problem of optimizing the linear programming bound for \(\Delta _d\) already appears somewhat similar to Problem 1.1, but there is a deeper analogy based on a problem studied by Cohn and Elkies in [5, Section 7]. Given an auxiliary function *f* for the sphere packing bound, let \(g = \widehat{f} - f\). Note that *g* is not identically zero, because otherwise *f* and \(\widehat{f}\) would both have compact support (thanks to their opposite signs outside radius *r*(*f*)), which would imply that \(f = \widehat{f} = 0\). Then *g* satisfies the conditions of the following problem, with \(r(g) \le r(f)\):

### Problem 1.3

*r*(

*g*) over all \(g :\mathbb {R}^d \rightarrow \mathbb {R}\) such that

- (1)
\(g\in L^1(\mathbb {R}^d){\setminus }\{0\}\) and \(\widehat{g} = -g\), and

- (2)
\(g(0)=0\) and

*g*is eventually nonnegative.

This problem has been solved for \(d \in \{1, 8, 24\}\), as a consequence of the sphere packing bounds mentioned above; the answers are 1, \(\sqrt{2}\), and 2, respectively. When \(d=2\), it is conjectured that the optimal value of *r*(*g*) is \((4/3)^{1/4}\), but no proof is known. No other closed forms have been identified.

Cohn and Elkies conjectured [5, Conjecture 7.2] that the minimal value of *r*(*g*) in Problem 1.3 is exactly the same as that of *r*(*f*) in the linear programming bound, and that in fact an auxiliary function *f* for the linear programming bound can always be reconstructed from an optimal *g* via \(g = \widehat{f} - f\). Nobody has proved that such an *f* always exists, but numerical evidence strongly supports this conjecture.

- (1)
\(f\in L^1(\mathbb {R}^d)\), \(\widehat{f}\in L^1(\mathbb {R}^d)\), and \(\widehat{f}\) is real-valued (i.e.,

*f*is even), - (2)
*f*is eventually nonnegative while \(\widehat{f}(0)\le 0\), and - (3)
\(\widehat{f}\) is eventually nonpositive while \(f(0)\ge 0\).

*g*in Problem 1.3 satisfies \(r(g) \ge \mathrm {A}_-(d)\).

For completeness, we state our next theorem for both \(\pm 1\) cases, although all the results in the following theorem were already proved for the \(+\,1\) case by Gonçalves, Oliveira e Silva, and Steinerberger in [12]. Note that we regard \(\mathrm {A}_{+\,1}\) and \(\mathrm {A}_{-1}\) as synonymous with \(\mathrm {A}_+\) and \(\mathrm {A}_-\), respectively.

### Theorem 1.4

*c*and

*C*such that

*d*. Moreover, for each

*d*there exists a radial function \(f\in \mathcal {A}_s(d){\setminus }\{0\}\) with \(\widehat{f} = s f\), \(f(0)=0\), and

In particular, \(\mathrm {A}_-(d) > 0\). Thus, we obtain a natural counterpart to the uncertainty principle of Bourgain, Clozel, and Kahane, but with *f* and \(\widehat{f}\) having opposite signs, and with the optimal function coming from Problem 1.3. We can take \(c = 1/\sqrt{2\pi e}\) and \(C=1\).

This uncertainty principle places the linear programming bound in a broader analytic context and gives a deeper significance to the auxiliary functions that optimize this bound. Outside of a few exceptional dimensions, they do not seem to come close to solving the sphere packing problem, but they conjecturally achieve an optimal tradeoff between sign conditions in the uncertainty principle.

Except for extremal functions for \(\mathrm {A}_+(1)\), our proof in Sect. 3.3 and the proof in [12] actually show that any extremal function cannot be eventually positive; that is, it must vanish on spheres with arbitrarily large radii, not just at infinitely many radii greater than \(\mathrm {A}_s(d)\). We strongly believe that this is the case for \(\mathrm {A}_+(1)\) as well.

Problems 1.1 and 1.3 are closely related and behave in complementary ways. We prove Theorem 1.4 by adapting the techniques of [12] to \(-\,1\) eigenfunctions. However, the analogy between these problems is not perfect. For example, the equality \(\mathrm {A}_+(12)=\mathrm {A}_-(8)=\sqrt{2}\) suggests that perhaps \(\mathrm {A}_+(28) = \mathrm {A}_-(24) = 2\), but that turns out to be false (see Sect. 4). Similarly, relatively simple explicit formulas show that \(\mathrm {A}_-(1)=1\), while \(\mathrm {A}_+(1)\) remains a mystery.

### Conjecture 1.5

See Sect. 4 for the numerical evidence supporting this conjecture. We expect that the common value of these limits is strictly between the bounds \(0.2419\ldots \) and \(0.3194\cdots \), and perhaps not so far from the latter.

In the remainder of the paper, we prove Theorem 1.2 in Sect. 2 and Theorem 1.4 in Sect. 3. In Sect. 4 we present numerical computations and conjectures, and we conclude in Sect. 5 with a construction of summation formulas that validate our numerics and lend support to our general conjectures about \(\mathrm {A}_s(d)\).

## 2 The \(+\,1\) eigenfunction uncertainty principle in dimension 12

In this section, we prove Theorem 1.2.

### 2.1 Optimality

We begin by establishing that \(\mathrm {A}_+(12) \ge \sqrt{2}\). For this inequality, we use a special Poisson-type summation formula for radial Schwartz functions \(f:\mathbb {R}^{12}\rightarrow \mathbb {C}\) based on the modular form \(E_6\). Converting a modular form into such a formula is a standard technique; for completeness, we will give a direct proof.

*j*. In particular, \(c_j{\!} >0\) for \(j\ge 1\) and we have the trivial bound \(c_j{\!}\le 504j^6\). Because \(E_6\) is a modular form of weight 6 for \(\mathrm {SL}_2(\mathbb {Z})\), it satisfies the identity

*f*(

*x*) with \(|x|=\sqrt{2j}\). Hence combining (2.1) and (2.2) yields

### Lemma 2.1

We follow the approach used to prove Theorem 1 in [16, Section 6].

### Proof

*F*, and note that \(\widehat{F}\) is also rapidly decreasing. By Fourier inversion,

*f*in the Schwartz topology. Moreover,

*L*is the \(D_{12}\) root lattice rescaled by a factor of \(1/\sqrt{2}\). Then the summation formula from Lemma 2.1 becomes a linear combination of the Poisson summation formulas for the lattices \(D_{12}\), \(D_{12}^*\),

*L*, and \(L^*\), which implies that it holds for all radial Schwartz functions. This argument shows that Lemma 2.1 is closely related to Poisson summation, while the proof we gave above applies directly to other modular forms as well as \(E_6\).

### Lemma 2.2

Let \(f \in \mathcal {A}_+(12)\). If both *r*(*f*) and \(r(\widehat{f}\,)\) are at most \(\sqrt{2}\), then \(f(x) = \widehat{f}(x) = 0\) whenever \(|x| = \sqrt{2j}\) with *j* a nonnegative integer.

### Proof

Without loss of generality, we can assume *f* is a radial function; otherwise, we simply average its rotations about the origin. (If the averaged function vanishes at radius \(\sqrt{2j}\), then so does *f* because \(r(f) \le \sqrt{2}\), and the same holds for \(\widehat{f}\).)

*f*is a radial Schwartz function, then Lemma 2.1 implies that

For general *f*, we can apply a standard mollification argument. Let \(\varphi :\mathbb {R}^d \rightarrow \mathbb {R}\) be a nonnegative, radial \(C^\infty \) function supported in the unit ball \(B^{d}_1\) with \(\widehat{\varphi }\ge 0\) and \(\widehat{\varphi }(0)=1\), so that the functions \(\varphi _\varepsilon \) defined for \(\varepsilon >0\) by \(\varphi _\varepsilon (x) = \varepsilon ^{-d}\varphi (x/\varepsilon )\) form an approximate identity.

Now let \(f_\varepsilon = (f * \varphi _\varepsilon ) \widehat{\varphi }_\varepsilon \). Because *f* and \(\widehat{f}\) are continuous functions that vanish at infinity, \(f_\varepsilon \rightarrow f\) and \(\widehat{f}_\varepsilon \rightarrow \widehat{f}\) uniformly on \(\mathbb {R}^d\) as \(\varepsilon \rightarrow 0\). Since \({{\,\mathrm{\mathrm {supp}}\,}}(\varphi _\varepsilon ) \subseteq B^{d}_\varepsilon \), we obtain the inequality \(f_\varepsilon (x) \ge 0\) whenever \(|x|\ge r(f) + \varepsilon \). Similarly \(\widehat{f_\varepsilon } = (\widehat{f} \ \widehat{\varphi }_\varepsilon )*\varphi _\varepsilon \), which implies that \(\widehat{f}_\varepsilon (x)\ge 0\) whenever \(|x|\ge r(\widehat{f}\,)+\varepsilon \). Furthermore, \(f_\varepsilon \) is a Schwartz function. To see why, note that \(\widehat{\varphi }_\varepsilon \) is a Schwartz function, while \(f * \varphi _\varepsilon \) is smooth and all its derivatives are bounded.

*f*, we again apply Lemma 2.1 to obtain

We will now apply this lemma to prove the lower bound \(\mathrm {A}_+(12) \ge \sqrt{2}\).

### Lemma 2.3

Suppose \(f \in \mathcal {A}_+(12)\). If \(r(f)r(\widehat{f}\,) < 2\), then *f* vanishes identically.

### Proof

By rescaling the input to *f*, we can assume without loss of generality that *r*(*f*) and \(r(\widehat{f}\,)\) are both less than \(\sqrt{2}\). Now we apply Lemma 2.2 to a rescaled version of *f*. Choose \(\lambda >0\) and let \(g(x) = f(\lambda x)\). Then \(\widehat{g}(\xi ) = \lambda ^{-12} \widehat{f}(\xi /\lambda )\), and it follows that \(g \in \mathcal {A}_+(12)\). Moreover, if \(\lambda \) is close enough to 1, then *r*(*g*) and \(r(\widehat{g})\) are both less than \(\sqrt{2}\).

*f*and \(\widehat{f}\) both have compact support, which implies that \(f=0\). \(\square \)

Exactly the same technique applies to any dimension and sign:

### Proposition 2.4

For example, for \(k\ge 2\), the summation formula coming from the Eisenstein series \(E_{2k}\) proves that \(\mathrm {A}_{(-1)^{k-1}}(4k) \ge \sqrt{2}\). This lower bound is sharp for \(k=2\) and \(k=3\), but it is not even true for \(k=1\), because \(E_2\) is merely a quasimodular form.

### Conjecture 2.5

For each \(s = \pm 1\) and \(d \ge 1\) except perhaps \((s,d)=(1,1)\), there is a summation formula that proves a sharp lower bound for \(\mathrm {A}_s(d)\) via Proposition 2.4.

In the case \(s=-\,1\), this conjecture is analogous to [3, Conjecture 4.2]. It holds in every case in which \(\mathrm {A}_s(d)\) is known exactly: the summation formulas that establish sharp lower bounds for \(\mathrm {A}_{-}(1)\), \(\mathrm {A}_{-}(8)\), and \(\mathrm {A}_{-}(24)\) are Poisson summation over the \(\mathbb {Z}\), \(E_8\), and Leech lattices, respectively, while the \(\mathrm {A}_{+}(12)\) case is Lemma 2.1. The conjectured value of \(\mathrm {A}_{-}(2)\) corresponds to Poisson summation over the isodual scaling of the \(A_2\) lattice. Conjecture 2.5 is not known to hold in any other case, nor can we guess what the summation formula should be, but the numerical and theoretical evidence in favor of this conjecture is compelling (see Sects. 4 and 5). In particular, in most cases we can compute the constants \(c_j{\!}\) and \(\rho _j\) in these conjectural summation formulas to high precision.

Summation formula that would prove \(\mathrm {A}_{+}(28) \ge 1.985406934891049\ldots \)

| \(\rho _j\) | \(c_j{\!}\) |
---|---|---|

0 | \(1.985406934891049\ldots \) | \(173693.2739265496\ldots \) |

1 | \(2.448204775489784\ldots \) | \(38022505.25862595\ldots \) |

2 | \(2.828451453989980\ldots \) | \(1612404204.870089\ldots \) |

3 | \(3.162301096885930\ldots \) | \(29295881893.82392\ldots \) |

4 | \(3.464102777388629\ldots \) | \(313503500519.3102\ldots \) |

5 | \(3.741654846843136\ldots \) | \(2325238355388.562\ldots \) |

6 | \(3.999999847797149\ldots \) | \(13196060863066.90\ldots \) |

\(\vdots \) | \(\vdots \) | \(\vdots \) |

We conjecture that there exists a formula of the form (2.3) in \(\mathbb {R}^{28}\) that agrees with all the digits listed in this table and proves a sharp lower bound for \(\mathrm {A}_{+}(28).\) |

### 2.2 Theta series and an extremal function in dimension 12

To prove the upper bound \(\mathrm {A}_+(12) \le \sqrt{2}\), we will construct an explicit function \(f \in \mathcal {A}_+(12)\) satisfying \(\widehat{f} = f\), \(f(0)=0\), and \(r(f) = \sqrt{2}\). To do so, we will use a remarkable integral transform discovered by Viazovska that turns modular forms into radial eigenfunctions of the Fourier transform. See [19] for background on modular forms, and [8, 9, 18] for other applications of this transform.

Viazovska’s method can be summarized by the following proposition, which is implicit in [18] but was stated there only for a specific modular form with \(d=8\) (and similarly for \(d=24\) in [8]). We omit the proof, because it closely follows the same approach as [18, Propositions 5 and 6] and [8, Lemma 3.1]. All that needs to be checked is the dependence on the dimension *d*.

### Proposition 2.6

*d*be a positive multiple of 4, and let \(\psi \) be a weakly holomorphic modular form of weight \(2-d/2\) for \(\Gamma (2)\) such that

*z*in the upper half-plane, \(t^{d/2-2}\psi (i/t) \rightarrow 0\) as \(t \rightarrow \infty \), and \(|\psi (it)| = O\big (e^{K\pi t}\big )\) as \(t \rightarrow \infty \) for some constant

*K*. Define a radial function \(f :\mathbb {R}^d \rightarrow \mathbb {R}\) by

*f*is a Schwartz function and an eigenfunction of the Fourier transform with eigenvalue \((-1)^{1+d/4}\). Furthermore,

Viazovska in fact developed two such techniques, one for each eigenvalue, and both are used in the sphere packing papers [8, 18]. We will not need the other technique, which yields eigenvalue \((-1)^{d/4}\) instead of \((-1)^{1+d/4}\) and uses a weakly holomorphic quasimodular form of weight \(4-d/2\) and depth 2 for \(\mathrm {SL}_2(\mathbb {Z})\).

*f*is a radial Schwartz function satisfying \(\widehat{f} = f\) and

For comparison, the quasimodular form inequalities that play the same role as (2.9) in [18] and [8] are obtained via computer-assisted proofs. The reason for this discrepancy is that those proofs combine \(+\,1\) and \(-\,1\) eigenfunctions, which introduces technical difficulties. If all one wishes to prove is that \(\mathrm {A}_-(8) = \sqrt{2}\) and \(\mathrm {A}_-(24)=2\), then one can avoid computer assistance. Specifically, the formula (3.1) in [8] is visibly positive in the same sense as our formula (2.6), and while that is not true for formula (46) in [18], it can be rewritten so as to be visibly positive (see, for example, the corresponding formula in [4]).

*f*(

*x*) with \(0 \le |x| \le \sqrt{2}\), we can simply cancel the growth of \(\psi (it)\). The series (2.7) shows that

*x*. It follows from (2.5) that

*f*(

*x*) is a holomorphic function of |

*x*|; thus, the new formula must agree with the old one for all

*x*by analytic continuation.

*f*(

*x*) must agree with

*f*(

*x*) has a single root at \(|x|=\sqrt{2}\) and a double root at the origin. More specifically,

In particular, \(f(0)=0\). It follows that \(f \in \mathcal {A}_+(12)\), and therefore \(\mathrm {A}_+(12) \le \sqrt{2}\), as desired. We have now proved all of the assertions from Theorem 1.2.

As the quadratic term \(-\,66\pi |x|^2\) suggests, our construction of *f* is scaled so that its values are rather large. For example, its minimum value appears to be \(f(x) \approx -\,23.8088\), achieved when \(|x| \approx 0.557391\). In Fig. 1, we have plotted a more moderate scaling of this function.

To arrive at the definition (2.6) of \(\psi \), we began with the Ansatz that \(\psi \Delta \) should be a holomorphic modular form of weight 8 for \(\Gamma (2)\). Equivalently, it should be a linear combination of \(\Theta _{00}^{16}\), \(\Theta _{00}^{12} \Theta _{01}^4\), \(\Theta _{00}^8 \Theta _{01}^8\), \(\Theta _{00}^4 \Theta _{01}^{12}\), and \(\Theta _{01}^{16}\). Imposing the constraint \(z^{4}\psi (-1/z) + \psi (z+1) = \psi (z)\) eliminates three degrees of freedom, which leaves just one degree of freedom, up to scaling. The remaining constraint is that the coefficient of \(e^{-\pi i z}\) in the Fourier expansion of \(\psi (z)\) must vanish, and then \(\psi \) is determined modulo scaling. Finally, we rewrote the formula for \(\psi \) to make it visibly positive.

## 3 The \(-\,1\) eigenfunction uncertainty principle

This section is devoted to the proof of Theorem 1.4. We deal only with the \(-1\) case, because all the assertions in this theorem were already proved in [12] for the \(+1\) case. First, we reduce determining \(\mathrm {A}_-(d)\) to solving Problem 1.3.

### Lemma 3.1

For each \(f\in \mathcal {A}_-(d){\setminus }\{0\}\), there exists a radial function \(g\in \mathcal {A}_-(d){\setminus }\{0\}\) such that \(\widehat{g} =-g\), \(g(0)=0\), and \(r(g) \le \sqrt{r(f)r(\widehat{f}\,)^{\phantom {\frac{0}{.}}}}\).

### Proof

If *f* is not radial, then we average its rotations about the origin to obtain a radial function without increasing *r*(*f*) or \(r(\widehat{f}\,)\). Thus, we can assume that *f* is radial. Note that this process cannot lead to the zero function: if it did, then *f* and \(\widehat{f}\) would both have compact support and hence vanish identically.

The quantity \(r(f)r(\widehat{f}\,)\) is unchanged if we replace *f* with \(x \mapsto f(\lambda x)\) for some \(\lambda >0\). Thus, we can assume that \(r(f)=r(\widehat{f}\,)\). Letting \(g=f-\widehat{f}\) we deduce that \(g\in \mathcal {A}_-(d)\), \(\widehat{g}=-g\), and \(r(g)\le r(f)\). Again, *g* cannot vanish identically, because *f* and \(-\widehat{f}\) are eventually nonnegative and would thus both have to have compact support.

*x*, and thus

*h*is not the zero function. \(\square \)

### 3.1 Lower and upper bounds

To obtain a lower bound for \(\mathrm {A}_-(d)\), we follow [2, 12]. Let \(g\in \mathcal {A}_-(d){\setminus }\{0\}\) be a radial function satisfying \(\widehat{g}=-g\) and \(g(0)=0\), and assume without loss of generality that \(\Vert g\Vert _1 = 1\).

*d*-dimensional ball of radius

*r*(

*g*) and centered at the origin, because \(\{x\in \mathbb {R}^d:g(x)< 0\}\subseteq B^d_{r(g)}\). It follows that

*n*with parameter \(\nu >-1\). When \(\nu = d/2-1\), the functions \(\psi _n^\nu :\mathbb {R}^d \rightarrow \mathbb {R}\) defined by

*p*with any polynomial of bounded degree, in the following sense. For \(N \ge 3\) and \(s = \pm 1\), let \(\mathrm {A}_{s,N}(d)\) be the infimum of

*r*(

*g*) over all nonzero \(g :\mathbb {R}^d \rightarrow \mathbb {R}\) such that \(\widehat{g} = sg\), \(g(0)=0\), and

*g*is of the form

*p*is a polynomial of degree at most

*N*. (The restriction to \(N \ge 3\) ensures that such a function exists.)

### Conjecture 3.2

*f*constructed in [11] for the linear programming bound in high dimensions. If we set \(g = \widehat{f} - f\), then one can show that

### 3.2 Existence of extremizers

*n*. In particular, since \(\widehat{f}_n=-f_n\), we have

*f*. Thus, necessarily we have \(\widehat{f}=-f\) and \(r(f)\le \mathrm {A}_-(d)\). Since \(\Vert f_n\Vert _{\infty }\le \Vert \widehat{f}_n\Vert _{1} = \Vert f_n\Vert _{1}= 1\) and \(r(f_n)\) is decreasing, we can apply Fatou’s lemma for \(g_n=\mathbf {1}_{B^d_{r(f_1)}} + f_n\ge 0\) to deduce that \(f\in L^1(\mathbb {R}^d)\) and \(\widehat{f}(0)\le 0\). Hence, \(f(0)\ge 0\). We now use Jaming’s high-dimensional version [14] of Nazarov’s uncertainty principle [15] to deduce, exactly as in [12, Lemma 23], that there exists \(K<0\) such that for all

*n*,

*K*but has a simpler proof.) Fatou’s lemma implies that

*f*satisfies the same estimate, and hence is not identically zero. We conclude that \(f \in \mathcal {A}_-(d)\), \(\widehat{f} = -f\), and \(r(f)\le \mathrm {A}_-(d)\), and thus \(r(f) = \mathrm {A}_-(d)\). Finally, we must have \(f(0)=0\), since otherwise the proof of Lemma 3.1 would produce a better function.

### 3.3 Infinitely many roots

All that remains to prove is that the extremizers have infinitely many roots. The proof follows the ideas of [12, Section 6.2] for the \(+1\) case. If \(f\in \mathcal {A}_-(d)\) satisfies \(\widehat{f}=-f\) and \(f(0)=0\) and vanishes at only finitely many radii beyond *r*(*f*), then we find a perturbation function \(g\in \mathcal {A}_-(d)\) satisfying \(\widehat{g} = -g\) and \(g(0)=0\) such that \(r(f+\varepsilon g) < r(f)\) for small \(\varepsilon >0\); thus, *f* cannot be extremal. In [12], the construction of *g* varies between the cases \(d=1\) (using the Poincaré recurrence theorem) and \(d\ge 2\) (using a trick involving Laguerre polynomials). However, thanks to the Poisson summation formula, every extremal function \(f\in \mathcal {A}_-(1)\) with \(\widehat{f}=-f\) and \(f(0)=0\) must vanish at the integers. Thus, we only need to prove our assertion for \(d\ge 2\).

In fact, we will rule out the possibility that an extremizer *f* is eventually positive. Then applying this proof to the radialization of *f* will show that *f* must vanish on spheres of arbitrarily large radius. Thus, let \(f \in \mathcal {A}_-(d)\) be such that \(\widehat{f} = -f\), \(f(0)=0\), and \(f(x)>0\) for \(|x| \ge R\). We must show that \(r(f) > \mathrm {A}_-(d)\).

*g*, but it needs to be fixed at the origin without changing its eventual nonnegativity. To do so, let \(\nu =d/2-1\) and consider the function

As observed in [12], for \(d \ge 2\) the eigenfunctions \(\psi ^\nu _j/\psi ^\nu _j(0)\) converge to zero uniformly on all compact subsets of \(\mathbb {R}^d{\setminus }\{0\}\) as \(j \rightarrow \infty \); the proof amounts to Fejér’s asymptotic formula for Laguerre polynomials [17, Theorem 8.22.1]. Using this convergence, let *n* be large enough that \(g_n(x) > 0\) for \(|x| \in [r(f),R]\), and then choose \(R'\) so that \(g_n(x) > 0\) for \(|x| \ge R'\). Let \(m=\min \{|f(x)| : R\le |x| \le R'\}\), \(M=\max \{|g_{n}(x)|: x\in \mathbb {R}^d\}\), and \(0<\varepsilon <m/M\). Then the perturbation \(f_\varepsilon = f + \varepsilon g_n\) satisfies \(f_\varepsilon (x) > 0\) for \(|x| \ge r(f)\). Thus, \(r(f_\varepsilon ) < r(f)\), which means *f* cannot be extremal. This completes the proof of Theorem 1.4.

## 4 Numerical evidence

To explore how \(\mathrm {A}_+(d)\) behaves, we numerically optimized functions \(g :\mathbb {R}^d \rightarrow \mathbb {R}\) satisfying the conditions of Problem 1.1. Readers who wish to examine this data can obtain our numerical results from [6].

In our calculations we always choose *g* to be of the form \(g(x) = p(2\pi |x|^2) e^{-\pi |x|^2}\), where *p* is a polynomial in one variable of degree at most \(4k+2\), which means *p* has \(4k+2\) degrees of freedom modulo scaling. The constraint \(g(0)=0\) eliminates one degree of freedom, and one can check using the Laguerre eigenbasis that the constraint \(\widehat{g} = g\) eliminates \(2k+1\) degrees of freedom. To control the remaining 2*k* degrees of freedom, we specify *k* double roots at radii \(\rho _1< \dots < \rho _k\). We then attempt to choose the radii \(\rho _1,\dots ,\rho _k\) so as to minimize *r*(*g*). To do so, we iteratively optimize the choice of radii for successive values of *k*, by making an initial guess based on the previous value of *k* and then improving the guess using multivariate Newton’s method. Each choice of \(\rho _1,\dots ,\rho _k\) proves an upper bound for \(\mathrm {A}_+(d)\), and we hope to approximate \(\mathrm {A}_+(d)\) closely as *k* grows. (Note that if Conjecture 3.2 holds, then we cannot obtain improved bounds if *k* remains bounded for large *d*.) This method was first applied by Cohn and Elkies [5, Section 7] to \(\mathrm {A}_-(d)\), with a simpler optimization algorithm. Cohn and Kumar [7] replaced that algorithm with Newton’s method, and we made use of their implementation.

We have no guarantee that the numerical optimization will converge to even a local optimum for any given *d* and *k*, or that the resulting bounds will converge to \(\mathrm {A}_+(d)\) as \(k \rightarrow \infty \). Indeed, we quickly ran into problems when \(d \le 2\), and eventually for \(d=3\) and 4 as well, but for \(5 \le d \le 128\) we arrived at the global optimum for each \(k \le 64\). These calculations are what initially led us to believe that \(\mathrm {A}_+(12)=\sqrt{2}\).

^{1}Thus, they are genuine theorems, while our numerical assertions about summation formulas have not been rigorously proved.

Upper bounds for \(\mathrm {A}_+(d)\) and \(\mathrm {A}_-(d-4)\)

| \(\mathrm {A}_+(d)\) | \(\mathrm {A}_-(d-4)\) | | \(\mathrm {A}_+(d)\) | \(\mathrm {A}_-(d-4)\) |
---|---|---|---|---|---|

1 | 0.572990 | 17 | 1.619692 | 1.627509 | |

2 | 0.756207 | 18 | 1.657044 | 1.665874 | |

3 | 0.887864 | 19 | 1.693390 | 1.703115 | |

4 | 0.965953 | 20 | 1.728806 | 1.739328 | |

5 | 1.036454 | 1 | 21 | 1.763360 | 1.774593 |

6 | 1.101116 | 1.074570 | 22 | 1.797112 | 1.808982 |

7 | 1.161109 | 1.141962 | 23 | 1.830115 | 1.842559 |

8 | 1.217275 | 1.203808 | 24 | 1.862417 | 1.875378 |

9 | 1.270241 | 1.261244 | 25 | 1.894060 | 1.907490 |

10 | 1.320483 | 1.315083 | 26 | 1.925084 | 1.938938 |

11 | 1.368375 | 1.365923 | 27 | 1.955522 | 1.969763 |

12 | \(\sqrt{2}\) | \(\sqrt{2}\) | 28 | 1.985407 | 2 |

13 | 1.458239 | 1.460307 | 29 | 2.014769 | 2.029684 |

14 | 1.500647 | 1.504478 | 30 | 2.043633 | 2.058842 |

15 | 1.541603 | 1.546952 | 31 | 2.072024 | 2.087503 |

16 | 1.581246 | 1.587911 | 32 | 2.099965 | 2.115691 |

For \(d \le 2\) our numerical methods perform poorly, for the reasons described below. For \(d=3\) the bound for \(\mathrm {A}_+(d)\) in Table 2 is obtained using \(k=27\), and for \(d \ge 4\) we use \(k=32\). In particular, we deliberately use a smaller value of *k* than the limits of our computations for \(d \ge 4\), so that we can use data from larger *k* to estimate the rate of convergence. These computations suggest the following conjecture.

### Conjecture 4.1

For \(3 \le d \le 32\), the upper bounds for \(\mathrm {A}_+(d)\) and \(\mathrm {A}_{-}(d-4)\) in Table 2 are sharp, except for an error of at most 1 in the last decimal digit shown.

In each case with \(d \ge 3\), we can use a summation formula to check that we have found the optimal bound for the given values of *d* and *k*; we explain how this is done in Sect. 5. However, we do not know how quickly the bounds converge as \(k \rightarrow \infty \), or whether they indeed converge to \(\mathrm {A}_s(d)\) at all. Our confidence in Conjecture 4.1 comes from comparing the bounds for \(32 \le k \le 64\) when \(d \ge 5\). They seem to have converged to this number of digits, but of course we cannot rule out convergence to the wrong limit.

The approximation \(\mathrm {A}_+(d) \approx \mathrm {A}_-(d-4)\) and equality \(\mathrm {A}_+(12) = \mathrm {A}_-(8) = \sqrt{2}\) raise the question of whether the other exact values \(\mathrm {A}_-(1)=1\), \(\mathrm {A}_-(2) = (4/3)^{1/4}\) (conjecturally), and \(\mathrm {A}_-(24)=2\) are also mirrored by \(\mathrm {A}_+\). That turns out not to be the case: Table 2 strongly suggests that \(\mathrm {A}_+(5) > 1\) and \(\mathrm {A}_+(6) > (4/3)^{1/4}\), and it proves that \(\mathrm {A}_+(28) < 2\). The case of \(\mathrm {A}_+(28)\) is particularly disappointing, because it might have stood in the same relationship to \(\mathrm {A}_+(12)\) as the Leech lattice does to the \(E_8\) root lattice. We have found no case other than \(d=12\) for which we can guess the exact value of \(\mathrm {A}_+(d)\).

Approximations to \(r(g)^2, \rho _1^2, \rho _2^2, \dots , \rho _{31}^2\) when \(d=28\) and \(k=128\)

3.9418406971135 | 20.000001150214 | 35.999999987965 | 52.000000000234 |

5.9937066227310 | 21.999999768273 | 37.999999967198 | 54.000000000902 |

8.0001376275780 | 23.999999651853 | 40.000000012100 | 55.999999999543 |

10.000148227366 | 25.999999804782 | 42.000000017800 | 58.000000002140 |

12.000008052312 | 28.000000118205 | 43.999999995225 | 60.000000000589 |

13.999980992905 | 30.000000112036 | 46.000000002272 | 61.999999999086 |

15.999998782377 | 31.999999979813 | 48.000000000644 | 63.999999999805 |

18.000002092309 | 33.999999997483 | 49.999999993657 | 65.999999999746 |

We view these numbers as approximations to the squared radii for the roots of a function achieving \(\mathrm {A}_+(28)\). |

### Conjecture 4.2

There exists a radial Schwartz function \(g \in \mathcal {A}_+(28) {\setminus } \{0\}\) with \(\widehat{g} = g\), \(g(0)=0\), and \(r(g) = \mathrm {A}_+(28)\), and whose nonzero roots are at radii \(\sqrt{2j + o(1)}\) as \(j \rightarrow \infty \), starting with \(j=2\).

This pattern is reminiscent of [10, Section 7], as well as the behavior of \(\mathrm {A}_\pm (d)\) in other cases, but it is a particularly striking example. We expect that Conjecture 4.2 is true, but a weaker conjecture consistent with the data is that there exists some \(\varepsilon <1\) such that the squared radii are within \(\varepsilon \) of successive even integers.

For comparison, [8] constructs a function achieving \(\mathrm {A}_-(24)\) whose nonzero roots are exactly at \(\sqrt{2j}\) with \(j \ge 2\). Our best guess is that the function achieving \(\mathrm {A}_+(28)\) is given by a primary term that has these exact roots, plus one or more secondary terms that perturb the roots but do not substantially change them. If that is the case, then perhaps one can describe this function explicitly and thereby characterize \(\mathrm {A}_+(28)\) exactly. However, we have not been able to guess or derive such a formula.

Another mystery is the behavior of \(\mathrm {A}_+(d)\) for \(d \le 2\). In these dimensions we quickly run into cases in which the last sign change *r*(*g*) is not a continuous function of \(\rho _1,\dots ,\rho _k\) at the optimum, and this lack of continuity ruins our numerical algorithms. (Instead, we resort to linear programming, which is much slower.) Of course it is no surprise that the last sign change is discontinuous at some points, because a small perturbation of a polynomial can convert a double root to two single roots, or even create a new root if the degree increases. However, we do not expect this behavior to occur generically. In particular, it cannot occur if \(\deg (p)=4k+2\) and *g* has no double roots beyond the *k* double roots we have forced to occur.

When \(d=2\), even the case \(k=1\) is problematic. Specifically, one can check that the optimal value \(r(g) = \sqrt{2/\pi }\) is achieved by setting \(\rho _1 = \sqrt{3/\pi }\). As \(\rho _1\) approaches \(\sqrt{3/\pi }\) from the left, *r*(*g*) decreases towards \(\sqrt{2/\pi }\), but it increases towards infinity as \(\rho _1\) approaches \(\sqrt{3/\pi }\) from the right. This discontinuity occurs because the leading coefficient of the polynomial *p* vanishes when \(\rho _1 = \sqrt{3/\pi }\). The leading coefficient also vanishes at the best choices of \(\rho _1,\dots ,\rho _k\) we have found for \(2 \le k \le 4\), while the case \(k=5\) suffers from a different problem: the resulting polynomial has six double roots, rather than just five, and again the location of the last sign change is discontinuous.

When \(d=1\), there are no problems for \(k \le 2\), and the leading coefficient vanishes for \(k=3\). For \(k=4\), we find an extra double root, but there is no discontinuity when \(k=5\).

In Table 2 we have reported the bound using \(k=5\) for \(d\le 2\). We believe that we have approximated the true optima for \(k=5\), but the bounds almost certainly do not agree with \(\mathrm {A}_+(d)\) to the full six digits shown, unlike Conjecture 4.1.

We have not observed a discontinuity near the optimum in any other dimension. However, when \(d=3\) we cannot find a local optimum with \(k=28\), because the largest root tends to infinity in our calculations. Computations carried out by David de Laat indicate that the optimum occurs at a singularity and the resulting discontinuity is interfering with our algorithms. When \(d=4\) we run into a similar problem at \(k=36\). We do not know whether this phenomenon is limited to \(d \le 4\).

## 5 Summation formulas

We do not know how to obtain the hypothetical summation formulas described in Conjecture 2.5. Aside from \(\mathrm {A}_-(2)\) and the four cases that have been solved exactly (namely \(\mathrm {A}_-(1)\), \(\mathrm {A}_-(8)\), \(\mathrm {A}_+(12)\), and \(\mathrm {A}_-(24)\)), we have not found any summation formulas that come close to matching our upper bounds. However, in many cases we can compute optimal summation formulas for polynomials of a fixed degree. For \(d \ge 3\), these formulas show that we have found the optimal polynomials for each fixed *k* in our computations in Sect. 4, and we believe that when *k* is large they should approximate the ultimate summation formulas. For example, Table 1 is based on calculations with \(k=128\).

*p*of \(q_0,q_1,\dots ,q_{2k+1}\) that vanishes at 0 and minimizes

*r*(

*p*); using the function \(f(x) = p(2\pi |x|^2)e^{-\pi |x|^2}\), we conclude that \(\mathrm {A}_s(d) \le \sqrt{r(p)/(2\pi )}\), where

*r*(

*p*), rather than \(|x| \ge R\), because we care only about the right half-line.) To construct

*p*, we impose double roots at locations \(\rho _1,\dots ,\rho _k\), and then choose these locations so as to minimize \(\rho _0 := r(p)\). Note that in our notation here, \(\rho _i\) denotes what would have been called \(2\pi \rho _i^2\) in Sect. 4.

*p*is uniquely determined among linear combinations of \(q_0,\dots ,q_{2k+1}\) by the following conditions:

- (1)
\(p(0)=0\),

- (2)
\(p(\rho _i) = p'(\rho _i)=0\) for \(1 \le i \le k\), and

- (3)
the coefficient of \(q_{2k+1}\) is 1.

*p*has roots of order exactly 1 at \(\rho _0\) and exactly 2 at \(\rho _1,\dots ,\rho _k\), and no other real roots greater than \(\rho _0\). Finally, we assume that we have found a strict local minimum for

*r*(

*p*); in other words,

*r*(

*p*) increases if we perturb \(\rho _1,\dots ,\rho _k\).

Values of *k* for which we have numerically computed a local minimum and the corresponding summation formula to one hundred decimal places

| | | | | |
---|---|---|---|---|---|

1 | 1 | 1, 2, 5 | \(-1\) | 1 | 1–64 |

1 | 2 | – | \(-\,1\) | 2 | 1–64 |

1 | 3 | 1–27 | \(-\,1\) | 3 | 1–20, 26–31 |

1 | 4 | 1–35 | \(-\,1\) | 4–128 | 1–64 |

1 | 5–128 | 1–64 | |||

1 | 28 | 1–128 | |||

When \((s,d) = (1,1)\), (1, 2), (1, 3), (1, 4), or \((-1,3)\), we believe the next value of |

### Proposition 5.1

*g*of \(q_0,\dots ,q_{2k+1}\). Furthermore, \(c_0,\dots ,c_k\) are nonzero and have the same sign. If \(s=1\), then \(c_{k+1}\) is nonzero and has the opposite sign.

We prove this proposition below. It is a polynomial analogue of the summation formula (2.4) (with the Gaussian factors from the Laguerre eigenbasis implicitly incorporated into the coefficients \(c_i\)), and it is reminiscent of Gauss-Jacobi quadrature in that it holds on a \((2k+2)\)-dimensional space despite using only \(k+2\) coefficients.

### Corollary 5.2

Any linear combination *g* of \(q_0,\dots ,q_{2k+1}\) with \(g(0)=0\) and \(r(g) < \rho _0\) must vanish identically, and *p* is the unique linear combination achieving \(r(p) = \rho _0\), up to scaling.

In other words, although we have assumed only a strict local minimum for the last sign change among polynomials with *k* double roots, we have found the global minimum among polynomials with no such restriction. For example, when \(s=1\) and \(k=64\), we find that *p* is the best possible polynomial of degree at most \(4k+2=258\). This phenomenon not only certifies our numerics by establishing matching lower bounds, but also helps explain why our algorithms perform well: degeneracy is the only way to get stuck in a local optimum.

### Proof of Corollary 5.2

*g*is a linear combination of \(q_0,\dots ,q_{2k+1}\) with \(r(g) \le \rho _0\), \(g(0) = 0\), and \(g(z) \ge 0\) for large

*z*. By Proposition 5.1,

*g*must vanish at \(\rho _0,\dots ,\rho _k\), since \(c_0,\dots ,c_k\) are nonzero and have the same sign. Furthermore, \(\rho _1,\dots ,\rho _k\) must be roots of even order, since otherwise

*g*would change sign beyond

*r*(

*g*). However, we have assumed that the equations \(g(0)=0\), \(g(\rho _i)=0\), and \(g'(\rho _i)=0\) for \(1 \le i \le k\) determine

*g*up to scaling. Thus

*g*must be proportional to

*p*, and the only way to achieve \(r(g) < r(p)\) is if

*g*vanishes identically. \(\square \)

It will prove convenient to distinguish between \(\rho _1,\dots ,\rho _k\) and perturbations of these points. For that purpose, we fix \(\rho _1,\dots ,\rho _k\) as the values described above, while \(\widetilde{\rho }_1,\dots ,\widetilde{\rho }_k\) are variables taking values in some neighborhood of \(\rho _1,\dots ,\rho _k\).

We write \(\widetilde{\rho }= (\widetilde{\rho }_1,\dots ,\widetilde{\rho }_k)\) and \(\rho = (\rho _1,\dots ,\rho _k)\). When necessary to avoid confusion, we write \(M(\widetilde{\rho })\) for the matrix depending on \(\widetilde{\rho }\), \(\alpha (\widetilde{\rho })\) for the solution of \(M(\widetilde{\rho }) \alpha = v\) if \(M(\widetilde{\rho })\) is invertible, and \(p_{\widetilde{\rho }}\) for the corresponding linear combination \(\sum _{j=0}^{2k+1} \alpha _j q_j\) of \(q_0,\dots ,q_{2k+1}\). Thus, the polynomial *p* discussed above amounts to \(p_{\rho }\).

We have assumed that \(M(\rho )\) is invertible, which means that \(\alpha (\widetilde{\rho })\) and \(p_{\widetilde{\rho }}\) are smooth functions of \(\widetilde{\rho }\) defined on some neighborhood of \(\rho \). Because \(p_{\rho }\) has a single root at \(\rho _0\), \(p_{\widetilde{\rho }}\) has a single root at some smooth function \(\widetilde{\rho }_0\) of \(\widetilde{\rho }_1,\dots ,\widetilde{\rho }_k\) with \(\widetilde{\rho }_0(\rho ) = \rho _0\), by the implicit function theorem. We will always assume that \(\widetilde{\rho }\) is in a small enough neighborhood of \(\rho \) for this to be true. Furthermore, our assumptions so far imply that \(r(p_{\widetilde{\rho }}) = \widetilde{\rho }_0\) for \(\widetilde{\rho }\) in some neighborhood of \(\rho \), and again we restrict our attention to such a neighborhood.

### Lemma 5.3

The vectors \(\alpha (\rho )\) and \((\partial \alpha / \partial \widetilde{\rho }_i)(\rho )\) with \(1 \le i \le k\) are linearly independent.

### Proof

The vector \(\alpha \) has \(\alpha _{2k+1}=1\), while all the partial derivatives \(\partial \alpha / \partial \widetilde{\rho }_i\) vanish in that coordinate. Thus, it will suffice to show that the partial derivatives are linearly independent at \(\rho \), and because *M* is invertible, we can examine \(M (\partial \alpha / \partial \widetilde{\rho }_i)\) instead of \(\partial \alpha / \partial \widetilde{\rho }_i\).

*i*and \(k+i\), and the entries of \((\partial M / \partial \widetilde{\rho }_i) \alpha \) in those rows are \(p_{\widetilde{\rho }}'(\widetilde{\rho }_i)\) and \(p_{\widetilde{\rho }}''(\widetilde{\rho }_i)\), respectively. We have \(p_{\widetilde{\rho }}'(\widetilde{\rho }_i)=0\) by construction, but \(p_{\rho }''(\rho _i) \ne 0\). Thus, the vectors \((\partial M / \partial \widetilde{\rho }_i)(\rho )\, \alpha (\rho )\) are linearly independent, as desired. \(\square \)

### Lemma 5.4

*g*of \(q_0,\dots ,q_{2k+1}\).

This lemma differs from Proposition 5.1 in not asserting uniqueness or sign conditions for \(c_0,\dots ,c_{k+1}\).

### Proof

*T*. To prove that such a vector exists, we will show that \({{\,\mathrm{\mathrm {rank}}\,}}(T) < k+2\).

It will suffice to find \(k+1\) linearly independent vectors in the kernel of left multiplication by *T*, because \((2k+2)-(k+1) < k+2\). Those vectors will be \(\alpha (\rho )\) and \((\partial \alpha /\partial \widetilde{\rho }_i)(\rho )\) for \(1 \le i \le k\), which are linearly independent by Lemma 5.3. All that remains is to prove that they are in the kernel of *T*.

*T*, as desired. \(\square \)

### Proof of Proposition 5.1

By Lemma 5.4, a summation formula exists, and all that remains is to prove uniqueness and the sign conditions.

Because \(M(\rho )\) is nonsingular, the values *g*(0) and \(g(\rho _i)\) with \(1 \le i \le k\) can be chosen arbitrarily. Thus, the summation formula must be unique up to scaling, and the coefficient \(c_0\) of \(\rho _0\) cannot vanish.

Now let \(1 \le i \le k\), and let \(\widetilde{\rho }\) equal \(\rho \) except in the *i*-th coordinate, where \(\widetilde{\rho }_i = \rho _i+\varepsilon \) with \(\varepsilon >0\) small. Then \(p_{\widetilde{\rho }}(\rho _i)\) and \(p_{\widetilde{\rho }}(\rho _0)\) have opposite signs because \(r(p_{\widetilde{\rho }}) > r(p_{\rho })\), while \(p_{\widetilde{\rho }}\) vanishes at the rest of \(\rho _1,\dots ,\rho _k\). It follows from taking \(g = p_{\widetilde{\rho }}\) that \(c_i\) must be nonzero, with the same sign as \(c_0\).

When \(s=-\,1\), we conjecture that \(c_{k+1}\) always has the same sign as \(c_0,\dots ,c_k\). This conjecture holds for every case listed in Table 4.

## Footnotes

- 1.
The non-sharp cases from Table 2 are straightforward to check rigorously, while the inequality \(\mathrm {A}_+(28) < 1.98540693489105\) requires more work because it uses a higher-degree polynomial with more complicated coefficients. We have proved it using the techniques and code from Appendix A of [7].

## Notes

### Acknowledgements

We thank Noam Elkies and the anonymous referees for their helpful comments on the manuscript, and David de Laat for carrying out computations by a different method that clarified the behavior of our computations for \(d \le 4\).

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