Amoebas, Nonnegative Polynomials and Sums of Squares Supported on Circuits

We completely characterize sections of the cones of nonnegative polynomials, convex polynomials and sums of squares with polynomials supported on circuits, a genuine class of sparse polynomials. In particular, nonnegativity is characterized by an invariant, which can be immediately derived from the initial polynomial. Furthermore, nonnegativity of such polynomials $f$ coincides with solidness of the amoeba of $f$, i.e., the Log-absolute-value image of the algebraic variety $\mathcal{V}(f) \subset (\mathbb{C}^*)^n$ of $f$. These results generalize earlier works both in amoeba theory and real algebraic geometry by Fidalgo, Kovacec, Reznick, Theobald and de Wolff and solve an open problem by Reznick. They establish the first direct connection between amoeba theory and nonnegativity of real polynomials. Additionally, these statements yield a completely new class of nonnegativity certificates independent from sums of squares certificates.


Introduction
Forcing additional structure on polynomials often simplifies certain problems in theory and practice. One of the most prominent examples is given by sparse polynomials, which arise in different areas in mathematics. Exploiting sparsity in problems can reduce the complexity of solving hard problems. An important example is, given by sparse polynomial optimization problems, see [20]. In this paper, we consider sparse polynomials having a special structure in terms of their Newton polytopes and supports. More precisely, we look at polynomials f ∈ R[x] = R[x 1 , . . . , x n ], whose Newton polytopes are simplices and the supports are given by all the vertices of the simplices and one additional interior lattice point in the simplices. Such polynomials have exactly n + 2 monomials and can be regarded as supported on a circuit. Note that A ⊂ N n is called a circuit, if A is affinely dependent, but any proper subset of A is affinely independent, see [11]. We write these polynomials as where the Newton polytope ∆ = New(f ) = conv{α(0), . . . , α(n)} ⊂ R n is a lattice simplex, y ∈ int(∆), b j ∈ R >0 and c ∈ R * . We denote this class of polynomials as P y ∆ . In this setting, the goal of this paper is to connect and establish new results in two different areas of mathematics. Namely, we link amoeba theory with nonnegative polynomials and sums of squares. The theory of amoebas deals with images of varieties V(f ) ⊂ (C * ) n under the Log-absolute-value map Log | · | : (C * ) n → R n , (z 1 , . . . , z n ) → (log |z 1 |, . . . , log |z n |), (1.2) having their nature in complex algebraic geometry with applications in various mathematical subjects including complex analysis [10,11], the topology of real algebraic curves [24], dynamical systems [8], dimers / crystal shapes [18], and in particular with strong connections to tropical geometry, see [25,28]. The cones of nonnegative polynomials and sums of squares arise as central objects in convex algebraic geometry and polynomial optimization, see [3,21].
For both amoebas and nonnegative polynomials / sums of squares, work has been done for special configurations in the above setting. In [37], the authors give a characterization of the corresponding amoebas of such polynomials and in [9,31], the authors characterize questions of nonnegativity and sums of squares for very special coefficients and simplices in the above sparse setting. We aim to extend results in all of these papers and establish connections between them for polynomials f ∈ P y ∆ .
We call a lattice point α ∈ Z n even if every entry α j is even, i.e., α ∈ (2Z) n . We call an integral polytope even if all its vertices are even. Finally, we call a polynomial a sum of monomial squares if all monomials b α x α satisfy b α > 0 and α even.
For the remainder of this article we assume that every polytope is even unless it is explicitly stated otherwise. However, we will reemphasize this fact in key statements.
For f ∈ P y ∆ we define the circuit number Θ f as where the λ j are uniquely given by the convex combination n j=0 λ j α(j) = y, λ j ≥ 0, n j=0 λ j = 1. We show that every polynomial f ∈ P y ∆ is, up to an isomorphism on R n , completely characterized by the λ j and its circuit number Θ f .
Remember that we always have c ∈ R * by definition of P y ∆ . The case c = 0 implies that the polynomial f is a sum of monomial squares and hence always is nonnegative. This should be kept in mind when with slight abuse of notation c = 0 is a possible choice in some statements. We now formulate our main theorems. The first theorem stated here is a composition of Theorem 3.8 and the Corollaries 3.9, 3.11 and 4.3 in the article. Theorem 1.1. Let f ∈ P y ∆ and ∆ be an even simplex, i.e., α(j) ∈ (2N) n for all 0 ≤ j ≤ n. Then the following statements are equivalent.
(2) f is nonnegative. Furthermore, f is located on the boundary of the cone of nonnegative polynomials if and only if |c| = Θ f for y / ∈ (2N) n and c = −Θ f for y ∈ (2N) n . In these cases, f has at most 2 n real zeros all of which only differ in their signs.
Assume that furthermore n ≥ 2 and f is not a sum of monomial squares with c > 0. Then the following are equivalent.
Note in this context that an amoeba A(f ) of f ∈ P y ∆ is solid if and only if its complement has no bounded components. Note furthermore that since ∆ is an even simplex, f is a sum of monomial squares (and hence trivially nonnegative) if and only if c ≥ 0 and y ∈ (2N) n . Theorem 1.1 yields a very interesting relation between the structure of the amoebas of f ∈ P y ∆ and nonnegative polynomials f ∈ P y ∆ , which are both completely characterized by the circuit number. Furthermore, it generalizes amoeba theoretic results from [37].
A crucial observation for f ∈ P y ∆ is that nonnegativity of such f does not imply that f is a sum of squares. It is particularly interesting that the question whether f ∈ P y ∆ is a sum of squares or not depends on the lattice point configuration of the Newton polytope of f alone. We give a precise characterization of the nonnegative f ∈ P y ∆ which are additionally a sum of squares in Section 5, Theorem 5.2. Here, we present a rough version of the statement.
Informal Statement 1.2. Let f ∈ P y ∆ and ∆ be an even simplex. Let f be nonnegative. Then f is a sum of squares if and only if y is the midpoint of two even distinct lattice points contained in a particular subset of lattice points in ∆. In particular, this is independent of the choice of the coefficients b j , c.
Note that Theorems 1.1 and 1.2 generalize the main results in [9] and [31] and yield them as special instances. In Section 5 we will explain this relationship in more detail.
Based on these characterizations we define a new convex cone C n,2d : Definition 1.3. We define the set of sums of nonnegative circuit polynomials (SONC) as for some even lattice simplices ∆ i ⊂ R n .
It follows by construction that membership in the C n,2d cone serves as a nonnegativity certificate, see also Proposition 7.2.
. Then f is nonnegative if there exist µ i ≥ 0, g i ∈ C n,2d for In Section 7 we discuss the SONC cone in further detail. In Proposition 7.2 we show that the SONC cone and the SOS cone are not contained in each other for general n and d. Particularly, we also prove that the existence of a SONC decomposition is equivalent to nonnegativity of f if New(f ) is a simplex and there exists an orthant where all terms of f except for those corresponding to vertices have a negative sign (Corollary 7.5).
Finally, we prove the following result about convexity, see also Theorem 6.4. Theorem 1.5. Let n ≥ 2 and f ∈ P y ∆ where ∆ is an even simplex. Then f is not convex.
Recently, there is much interest in understanding the cone of convex polynomials, Theorem 1.5 serves as an indication that sparsity is a structure that can prevent polynomials from being convex.
Further contributions. 1. Gale duality is a standard concept for (convex) polytopes, matroids and sparse polynomial systems, see [2,11,14,35]. We show that a polynomial f ∈ P y ∆ has a global norm minimizer e s * ∈ R n , see Section 3.2. f at e s * together with the circuit number Θ f equals the Gale dual vector of the support matrix up to a scalar multiple (Corollary 3.10). Furthermore, it is an immediate consequence of our results that the circuit number is strongly related to the A-discriminant of f . Particularly, f ∈ P n,2d ∩ P y ∆ is contained in the topological boundary of the nonnegativity cone, i.e., f ∈ ∂(P n,2d ∩ P y ∆ ), if and only if the A-discriminant vanishes at f (Corollary 3.11). These facts about the A-discriminant were first shown in [26] and [37].
2. We consider the case of multiple interior lattice points in the support of f . We prove for the case that all coefficients of the interior monomials are negative that all such nonnegative polynomials are in C n,2d . Furthermore, we show when such polynomials are sums of squares, again generalizing results in [9].
3. Since the condition of being a sum of squares depends on the combinatorial structure of the simplex ∆, using techniques from toric geometry, we provide sufficient conditions for simplices ∆ such that every nonnegative polynomial in P y ∆ is a sum of squares, independent from the position of y ∈ int(∆). This will prove that for n = 2 almost every nonnegative polynomial in P y ∆ is a sum of squares and this also yields large sections on which nonnegative polynomials and sums of squares coincide.
4. We answer a question of Reznick stated in [31] whether a certain lattice point criterion on a class of sparse support sets (more general than circuits) of nonnegative polynomials is equivalent to these polynomials being sums of squares.
This article is organized as follows. In Section 2, we introduce some notations and recall some results that are essential for the upcoming sections and proofs of the main theorems. In Section 3, we characterize nonnegativity of polynomials f ∈ P y ∆ . This is done via a norm relaxation method, which is outlined in the beginning of the section. Furthermore, Section 3 deals with invariants and properties of such polynomials and sets them in relation to Gale duals and A-discriminants. In Section 4, we discuss amoebas of polynomials f ∈ P y ∆ and how they are related to nonnegativity respectively the circuit number. In Section 5 we completely characterize the section of the cone of sums of squares with P y ∆ . Furthermore, we generalize results regarding nonnegativity and sums of squares to non-sparse polynomials with simplex Newton polytope. In Section 6, we completely characterize convex polynomials in P y ∆ . In Section 7, we provide and discuss a new class of nonnegativity certificate given by sums of nonnegative circuit polynomials (SONC). In Section 8, we prove that for non-simplex Newton polytopes Q the lattice point criterion from the simplex case does not suffice to characterize sums of squares. We show that a necessary and sufficient criterion can be given by additionally taking into account the set of possible triangulations of Q. This solves an open problem stated by Reznick in [31]. Finally, in Section 9, we provide an outlook for future research possibilities.

Acknowledgments
We would like to thank Christian Haase for his support and explanations concerning toric ideals and normality. Furthermore, we thank Jens Forsgård, Hannah Markwig, Frank Sottile and Thorsten Theobald for their helpful comments and suggestions on the manuscript. Moreover, we thank the anonymous referees for their helpful comments.
and the convex cone of sums of squares as For an introduction of nonnegative polynomials and sums of squares, see [3,21,22]. Since we are interested in nonnegative polynomials and sums of squares in the class P y ∆ , we consider the sections P y n,2d = P n,2d ∩ P y ∆ and Σ y n,2d = Σ n,2d ∩ P y ∆ . 2.2. Amoebas. For a given Laurent polynomial f ∈ C[z 1 , . . . , z n ] on a support set A ⊂ Z n with variety V(f ) ⊂ (C * ) n , the amoeba A(f ) is defined as the image of V(f ) under the log-absolute map Log | · | defined in (1.2). Amoebas were first introduced by Gelfand, Kapranov and Zelevinsky in [11]. For an overview see [7,25,28,33].
Amoebas are closed sets [10]. Their complements consists of finitely many convex components [11]. Each component of the complement of A(f ) corresponds to a unique lattice point in conv(A) ∩ Z n via an order map [10].
Components of the complement which correspond to vertices of conv(A) via the order map do always exist. For all other components of the complement of an amoeba A(f ) the existence depends non trivially on the choice of the coefficients of f , see [11,25,28]. We denote the component of the complement of A(f ) of all points with order α ∈ conv(A)∩Z n as E α (f ).
The fiber F w of each point w ∈ R n with respect to the Log | · |-map is given by It is easy to see that F w is homeomorphic to a real n-torus (S 1 ) n . For f = α∈A b α z α and v ∈ (C * ) n we define the fiber function This means that f |v| is the pullback ϕ * |v| (f ) of f under the homeomorphism ϕ |v| : (S 1 ) n → F Log |v| ⊂ (C * ) n . The crucial fact about the fiber function is that for its zero set V(f |v| ) it holds that and hence we have for the amoeba A(f ) that For more details on the fiber function see [7,25,34,37].
2.3. Agiforms. Asking for nonnegativity of polynomials supported on a circuit is closely related objects called an agiform in [31]. Given a even lattice simplex ∆ ⊂ R n and an interior lattice point y ∈ int(∆), the corresponding agiform to ∆ and y is given by The term agiform is implied by the fact that the polynomial f (∆, λ, y) = n i=0 λ i x α(i) − x y is nonnegative by the arithmetic-geometric mean inequality. Note that an agiform has a zero at the all ones vector 1. This implies that agiforms lie on the boundary of the cone of nonnegative polynomials. A natural question is to characterize those agiforms that can be written as sums of squares. In [31], it is shown that this depends non-trivially and exclusively on the combinatorial structure of the simplex ∆ and the location of y in the interior. We need some definitions and results adapted from [31].
(2) We say that L is∆-mediated, if i.e., every β ∈ L \∆ is an average of two distinct even points in L.
The main result in [31] concerning the question under which conditions agiforms are sums of squares is given by the following theorem.

Invariants and Nonnegativity of Polynomials Supported on Circuits
The main contribution of this section is the characterization of P y n,2d , i.e., the set of nonnegative polynomials supported on a circuit (Theorem 3.8). Along the way we provide standard forms and invariants, which reflect the nice structural properties of the class P y ∆ . In Section 3.1 we outline the norm relaxation method, which is the proof method used for the characterization of nonnegativity. In Section 3.2, we introduce standard forms for polynomials in P y ∆ and, in particular, prove the existence of a particular norm minimizer for polynomials, where the coefficient c equals the negative circuit number Θ f (Proposition 3.4). In Section 3.3, we put all pieces together and characterize nonnegativity of polynomials in P y ∆ (Theorem 3.8). In Section 3.4, we discuss connections to Gale duals and A-discriminants.
3.1. Nonnegativity via Norm Relaxation. We start with a short outline of the proof method, which we introduce and apply here in order to tackle the problem of nonnegativity of polynomials. Let f = α∈A b α x α ∈ R[x] be a polynomial with A ⊂ N n finite, 0 ∈ A and α ∈ (2N) n as well as b α > 0 if α is contained in the vertex set vert(A) of conv(A). Instead of trying to answer the question whether f (x) ≥ 0 for all x ∈ R n , we investigate the relaxed problem Since the strict positive orthant R n >0 is an open dense set in R n ≥0 and the componentwise exponential function Exp : R n → R n >0 , (x 1 , . . . , x n ) → (exp(x 1 ), . . . , exp(x n )) is a bijection, Problem (3.1) is equivalent to the question Hence, an affirmative answer of (3.2) implies nonnegativity of f . The motivation for the relaxation is that, on the one hand, Question (3.2) is eventually easier to answer, since we have linear operations on the exponents and, on the other hand, the gap between (3.2) and nonnegativity hopefully is not too big, in particular for sparse polynomials. We show that for polynomials supported on a circuit (and some more general classes of sparse polynomials) both is true: In fact, for circuit polynomials the question of nonnegativity and (3.2) is equivalent and can be characterized exactly, explicitly, and easily in terms of the coefficients of f and the combinatorial structure of A.
An interesting side effect of the described relaxation is that (3.2) is strongly related to the amoeba of f as we point out (for circuit polynomials) in the following Section 4. Thus, it will serve us as a bridge between real algebraic geometry and amoeba theory.

Standard Forms and Norm Minimizers of Polynomials Supported on Cir-
cuits. Let f be a polynomial of the Form (1.1) defined on a circuit A = {α(0), . . . , α(n), y} ⊂ Z n . Observe that there exists a unique convex combination n j=0 λ j α(j) = y. In the following, we assume without loss of generality that α(0) = 0, which is possible, since we can factor out a monomial x α(0) with α(0) ∈ (2N) n if necessary. We define the support matrix M A by and M A j as the matrix obtained by deleting the j-th column of M A , where we start to count at 0. Furthermore, we always assume that b 0 = λ 0 , which is always possible, since multiplication with a positive scalar does not affect if a polynomial is nonnegative. We denote the canonical basis of R n with e 1 , . . . , e n . Proposition 3.1. Let f be of the Form (1.1) supported on a circuit A = {α(0), . . . , α(n), y} ⊂ Z n and y = n j=0 λ j α(j) with n j=0 λ j = 1, 0 < λ j < 1 for all j. Let µ ∈ N >0 denote the least common multiple of the denominators of the λ j . Then there exists a unique polynomial g of the Form (1.1) with supp(g) = A ′ = {0, α(1) ′ , . . . , α(n) ′ , y ′ } ⊂ Z n such that the following properties hold.
For every f of the Form (1.1) we call the polynomial g, which satisfies all the conditions of the proposition, the standard form of f . Note that f (e w ) is defined in the sense of (3.2) and the support matrix M A ′ of the standard form of f is of the shape Proof. We assume without loss of generality that α(0) = 0. Let M n+1 e j for 1 ≤ j ≤ n. We construct the polynomial g. We choose the same coefficients for g as for f . Since 0, α(1), . . . , α(n) form a simplex, there exists a unique matrix T ∈ GL n (Q) such that with M A ′ of the Form (3.3) given by µT = (M A n+1 ) −1 . Since y = n j=0 λ j α(j), it follows that, in affine coordinates, we have y ′ j e j = T −1 λ j (M A n+1 e j ), i.e., y ′ = µ(λ 0 , . . . , λ n ). Thus, (1) -(4) holds.
We show that f (e w ) = g(e T t w ) for every w ∈ R n . We investigate the monomial x α(j) : For the inner monomials y and y ′ we know that y = T y ′ and thus for y ′ = n j=0 λ j α(j) ′ we have y = T ( n j=0 λ j α(j) ′ ) = n j=0 λ j T α(j) ′ = n j=0 λ j α(j). Therefore, (5) follows from Proposition 3.1 can easily be generalized to polynomials with New(f ) = ∆ = conv{0, α(1), . . . , α(n)} being a simplex and I ⊂ (int(∆) ∩ Z n ). Every y(i) has a unique convex combination y(i) = λ Proof. By definition of µ, the support matrix M A ′ is integral again. Since in the proof of Proposition 3.1 neither uniqueness of y is used nor special assumptions about y were made, the statement follows. Now, we return to the case of circuit polynomials.
with w ∈ R n has a unique extremal point, which is always a minimum.
This proposition was used in [37] (see Lemma 4.2 and Theorem 5.4). For convenience, we give an own, easier proof here.
Proof. We investigate the standard form g of f . For the partial derivative x j ∂g/∂x j (we can multiply with x j , since e w ≥ 0) we have Hence, the partial derivative vanishes for some e w if and only if Since the right hand side is strictly positive, we can apply log | · | on both sides for every partial derivative and obtain the following linear system of equations  Since the matrix on the left hand side has full rank, we have a unique solution.
For arbitrary f we have f (e w ) = g(e T t w ) by Proposition 3.1 and, hence, if w * is the unique extremal point for g(e w ), then (T t ) −1 w * is the unique extremal point for f (e w ).
For every w ∈ R n with ||w|| → ∞ the polynomial f converges against the terms with exponents which are contained in a particular proper face of New(f ). Since all these terms are strictly positive, f (e w ) converges against a number in R >0 ∪ {∞}. Thus, the unique extremal point has to be a global minimum.
For f ∈ P y ∆ we define s * f ∈ R n as the unique vector satisfying s * f indeed is well defined, since application of log | · | on both sides yields a linear system of equations with variables s * k,f and the rank of this system has to be n, since conv(A) is a simplex. If the context is clear, then we simply write s * instead of s * f and e s * instead of e s * f . We recall that the circuit number associated to a polynomial f ∈ P y ∆ is given by is a root and the unique global minimizer of f (e w ).
Due to this proposition we call the point s * the norm minimizer of f . We remark that this proposition was already shown for polynomials in P y ∆ in standard form in [9] and for arbitrary simplices but in a more complicated way in [37].
Proof. For f (e s * ) we have For the minimizer statement, we investigate the partial derivatives x j ∂f /∂x j (we can multiply with x j , since e w > 0). Since y j = n k=1 λ j α j (k), we obtain Evaluation of the partial derivative at e s * yields Finally, by Proposition 3.3, e s * is the unique global minimizer of f (e w ).
In some contexts it is more convenient to work with a Laurent polynomial supported on a circuit where the interior point y equals the origin. With the same argumentation as before we find a suitable standard form.
(2) f and g have the same coefficients, For every polynomial f of the Form (1.1), we call the polynomial g, which satisfies all conditions in Corollary 3.5, the zero standard form of f . Note that the support matrix M A ′′ of the zero standard form of f is of the shape Proof. We divide f by x y , which is always possible, since e w > 0. We apply literally the proof of Proposition 3.1 with the exception of using the matrix M A 0 instead of M A n+1 and the convex combination −λ 0 α(0) = n j=1 λ j α(j) instead of y = n j=0 λ j α(j). An advantage of the zero standard form is that the global minimizer does not longer depend on the choice of c. Corollary 3.6. For f ∈ P y ∆ the point e s * is a global minimizer for (f /x y )(e w ) independent of the choice of c.
Proof. By Corollary 3.5, we can transform f into zero standard form with y = 0. Then the proof of Proposition 3.4 can be literally applied again with the exception of (f /x y )(e w ) = 0 if and only if c = −Θ f .

3.3.
Nonnegativity of Polynomials Supported on a Circuit. In this section, we characterize nonnegativity of polynomials in P y ∆ . The following lemma allows us to reduce the case of y ∈ ∂∆ to the case y ∈ int(∆).
x y be such that the Newton polytope is given by ∆ = New(f ) = conv{0, α(1), . . . , α(n)} and y ∈ ∂∆. Furthermore, let F be the face of ∆ containing y. Then f is nonnegative if and only if the restriction of f to the face F is nonnegative.
Proof. For the necessity of nonnegativity of the restricted polynomial, see [31]. Otherwise, the restriction to the face F contains the monomial x y and this restriction is nonnegative. Since all other terms in f correspond to the (even) vertices of ∆ and have nonnegative coefficients, the claim follows. Now, we show the first part of our main Theorem 1.1 by characterizing nonnegative polynomials f ∈ P y ∆ supported on a circuit. Recall that we denote such polynomials of degree 2d in in n variables as P y n,2d . Note that this theorem covers the known special cases of agiforms [31] and circuit polynomials in standard form [9].
Then the following are equivalent.
Proof. First, observe that f ≥ 0 is trivial for c ≥ 0 and y ∈ (2N) n , since in this case f is a sum of monomial squares. We apply the norm relaxation strategy introduced in Section 3.1. Initially, we show that f (x) ≥ 0 if and only if f (e w ) ≥ 0 for all f ∈ P y ∆ . Let without loss of generality y 1 , . . . , y k be the odd entries of the exponent vector y. Thus, for every 1 ≤ j ≤ k replacing x j by −x j changes the sign of the term c · x y . Since all other terms of f are nonnegative for every choice of x ∈ R n , we have f (x) ≥ 0 if sgn(c) · sgn(x 1 ) · · · sgn(x k ) = 1. Since furthermore, for sgn(c) · sgn(x 1 ) · · · sgn(x k ) = −1 we have c · x y = −|c| · |x 1 | y 1 · · · |x n | yn , we can assume c ≤ 0 and x ≥ 0 without loss of generality. Then λ 0 + n j=0 b j x α(j) − |c||x| y is nonnegative for all x ∈ R n if and only if this is the case for all x ∈ R n ≥0 . And since R n >0 is an open, dense set in R n ≥0 , we can restrict ourselves to the strict positive orthant. With the componentwise bijection between R n >0 and R n given by the Exp-map, it follows that f (x) ≥ 0 for all x ∈ R n if and only if f (e w ) ≥ 0 for all w ∈ R n . Hence, the theorem is shown if we prove thatf (e w ) ≥ 0 for all w ∈ R n if and only if c ∈ [−Θ f , 0]. We fix some arbitrary b 1 , . . . , b n ∈ R >0 and denote by (f c ) c∈R be the corresponding family of polynomials in P y ∆ . By Proposition 3.4, f c (e w ) has a unique global minimum But this fact also completes the proof for general c < 0: Since c · e w,y is the unique negative term in f c (e w ) for all w ∈ R n , a term by term inspection yields that An immediate consequence of the theorem is an upper bound for the number of zeros of polynomials f ∈ ∂P y n,2d . Corollary 3.9. Let f ∈ ∂P y n,2d . Then f has at most 2 n affine real zeros v ∈ R n , which all satisfy |x j | = e s * j for all 1 ≤ j ≤ n. Proof. Assume f ∈ ∂P y n,2d and f (x) = 0 for some x ∈ R n . Then we know by the proof of Theorem 3.8 that |x j | = e s * j . Thus, x = (±e s * 1 , . . . , ±e s * n ). The bound in Corollary 3.9 is sharp as demonstrated by the well-known Motzkin poly- 6 . The zeros are given by x = (±1, ±1). Furthermore, it is important to note that the maximum number of zeros does not depend on the degree of the polynomials, which is in sharp contrast to previously known results concerning the maximum number of zeros of nonnegative polynomials and sums of squares, [6].
In order to illustrate the results of this section, we give an example. Let f = 1 + . We apply Proposition 3.1 and compute the standard form g of 1/3 · f . Then g is the polynomial, which is supported on a circuit Additionally, g has the same coefficients as f . It is easy to see that and thus Since the circuit number Θ f only depends on the coefficients of f and the convex combination of y, it is invariant with respect to transformation to the standard form. Thus, we have But the inner coefficient c of the Motzkin polynomial equals its negative circuit number. Hence, the Motzkin polynomial is contained in the boundary of the cone of nonnegative polynomials.
We give a second example where nonnegativity is not a priori known Again, it is easy to see that λ 1 = 1/2 and λ 2 = 1/4. Hence, And since |c| < Θ f , we can conclude that f is a strictly positive polynomial.

3.4.
A-Discriminants and Gale Duals. For a given (n + 1) × m support matrix M A with A ⊂ Z n and conv(A) being full dimensional, a Gale dual or Gale transformation is an integral m × (m − n − 1) matrix M B such that its rows span the Z-kernel of M A . In other words, for every integral vector v ∈ Z m with M A v = 0, it holds that v is an integral linear combination of the rows of M B , see [11,28]. If A is a circuit, then M B is a vector with n + 2 entries. It turns out that this vector is closely related to the global minimum e s * ∈ R n and the circuit number Θ f . Furthermore, we point out that the circuit number Θ f and the question of nonnegativity is closely related to A-discriminants. Let A = {α(1), . . . , α(d)} ⊂ Z n and let C A denote the space of all polynomials d j=1 b j z α(j) with b j ∈ C. Since every (Laurent-) polynomial in C A is uniquely determined by its coefficients, C A can be identified with a C d space. Let ∇ A be the Zariski closure of the subset of all polynomials f in C A for which there exists a point z ∈ (C * ) n such that which has the variety ∇ A , see [11].
The following statement is an immediate consequence of Proposition 3.4 and Theorem 3.8. But it was (at least implicitly) already known before and can also be derived from [11], [37], and [26].

Amoebas of Real Polynomials Supported on a Circuit
In this section, we investigate amoebas of real polynomials supported on a circuit. We show that for amoebas of polynomials of the Form (1.1), which are not a sum of monomial squares, a point w is contained in a bounded component of the complement only if the norm of the "inner" monomial is greater than the sum of all "outer" monomials at w ∈ R n (Theorem 4.2). This implies particularly that an amoeba of this type has a bounded component in the complement if and only if the "inner" coefficient c satisfies |c| > |Θ f |, which proves the equivalence of (1) and (2) in Theorem 1.1. Furthermore, this result generalizes some statements in [37].
In this section, we always assume that f c is a parametric family of a Laurent polynomial of the Form (1.1) with real parameter c ∈ R ≤0 . Furthermore, we always assume that f c is given in zero standard form (see Section 3), i.e., Let w ∈ R n be an arbitrary point in the underlying space of A(f c ). As introduced in Section 2.2, we denote the fiber with respect to the Log | · |-map as F w and the fiber function of f c at the fiber F w as f | exp(w)| c . We define the following parameters: |b j e w,α(j) |, The following facts about amoebas supported on a circuit are well-known.
1 , . . . , z ±1 n ] be a Laurent polynomial with b j ∈ C * and c ∈ C such that New(f ) is a simplex and y ∈ int(New(f )).
(1) The complement of A(f ) has exactly n + 1 unbounded and at most one bounded component. If the bounded component E y (f ) exists, then it has order y. Part (1) and (2) are consequences of a Theorem by Rullgård based on tropical geometry, which was applied to the circuit case by Theobald and the second author, see [37, Lemma 2.1] and also [7,Theorem 4.1]. Part (3) is an immediate consequence of Purbhoo's lopsidedness condition (also referred as generalized Pellet's Theorem), see [30]. Part (4) is [37,Theorem 4.4] after investigating f in the standard form introduced in Section 3, which guarantees that the bound given in [37,Theorem 4.4] coincides with the circuit number Θ f . Note that this means Θ f = min w∈R n Θ w . Similarly, we define Ψ f = min w∈R n Ψ w . We remark that Ψ f is the minimal choice for |c| such that the tropical hypersurface T (trop(f )) of the tropical polynomial trop(f ) = n+1 j=1 log |b j |⊙x α(j) ⊕log |c| has genus one, see [7,37] for details; for an introduction to tropical geometry see [23].   Proof. The corollary follows immediately from Theorem 4.1 (2) and (4), Theorem 4.2 (including its consecutive note) and the fact that Θ f = Θ s * = min w∈R n Θ w by Corollary 3.6.
The proof of Theorem 4.2 will be quite a lot of work. We need to show a couple of technical statements before we can tackle the actual proof. The first lemma which we need was similarly used in [37,Theorem 4.1].
For convenience, we provide the proof again. It is mainly based on the Rouché theorem. Recall that the winding number of a closed curve γ in the complex plane around a point z is given by 1 2πi γ dζ ζ−z . Proof. Assume |b 1 | > |b 2 |. Clearly, the function b 1 · e i·rφ has a non-zero winding number around the origin. If g would have a winding number of zero around the origin, then there would exist some t ∈ (0, 1) such that h(φ) = b 1 · e i·rφ + t · b 2 · e i·(η+sφ) has a zero φ outside the origin. This is a contradiction. Hence, the trajectory of g needs to intersect the real line in the strict positive part as well as in the strict negative part.
Since g is continuous in the norms of its coefficients, the statement can be extended to |b 1 | = |b 2 | and intersections of g with the nonnegative part as well as the nonpositive part of the real axis. Now, we step over to complex functions on the real n-torus (S 1 ) n .
At the endpoint γ(k j ) of the path segment [k j−1 , k j ] ⊂ [0, 1], we are therefore in the situation g(γ(k j )) ≤ |b 1 | + |b j+1 | + · · · + |b n−1 | + Re(b n ) + Re(b n+1 ) − j l=2 |b l |. We can glue together different path segments, since for each γ(k j ) we have φ 1 = 0 by construction and the value of φ n does not matter. Thus, we can subsequently repeat the procedure for all j until we reach j = n − 1 and obtain a complete path γ ⊂ (S 1 ) n with the desired properties.
For the next step of the proof we need to recall the definition of a hypotrochoid. A hypotrochoid with parameters R, r ∈ Q >0 , d ∈ R >0 satisfying R ≥ r is the plane algebraic curve γ in R 2 ∼ = C given by Geometrically, a hypotrochoid is given the following way: Let a small circle C 1 with radius r roll along the interior of a larger circle C 2 with radius R. Mark a point p at the end of a segment with length d starting at the center of C 1 . Then the hypotrochoid is the trajectory of p.
We say that a curve γ is a hypotrochoid up to a rotation if there exists some re- k is a hypotrochoid. If k = 0 or k = π, then we say that γ is a real hypotrochoid. Hypotrochoids are closed, continuous curves in the complex plane, which attain values in the closed annulus with outer radius (R − r) + d and inner radius (R − r) − d for (R − r) ≥ d. Furthermore, if they are real, then they are symmetric along the real line. For an overview about hypocycloids and other plane algebraic curves see [4].
In order to prove the second key lemma, which is needed for the proof of Theorem 4.2, we make use of the following special case of [38, Theorem 4.1]. Lemma 4.6. Let g : S 1 → C, φ → e isφ +qe −itφ +p with p, q ∈ C * . Then g is a hypotrochoid up to a rotation around the point p with parameters R = (t + s)/t, r = s/t and d = |q| rotated by arg(q) · s.

The proof of this lemma is a straightforward computation.
Proof. The non-constant part g − p of the function g is given by (g − p)(φ) = e isφ + |q| · e i·(arg(q)−tφ) .
Since, with our choice of parameters, R − r = 1 and (r − R)/r = −t/s, it follows by (4.2) after replacing φ by φ ′ = sφ that g − p is a hypotrochoid up to a rotation.
Lemma 4.6 about hypotrochoids allows us to prove the following technical lemma.
We investigate the fiber function We have to show that V(f Let now n ∈ {2, 3}. If n = 3, the we apply Lemma 4.5, fix φ 2 = π, and restrict f |1| c to which is defined on the sub 2-torus of F |1| given by (φ 1 , φ n ). Since (µ · λ 2 )/λ 0 is an integer, b n+1 · e −i·µ·λ 2 /λ 0 is real and hence we can apply Lemma 4.7. It yields that g(φ 1 , φ n ) c attains all real values in the interval We close the section with the proof of Lemma 4.7.
Proof. (Proof of Lemma 4.7) For every fixed value of φ 1 ∈ [0, 2π) the values of g are given by a curve of the form By Lemma 4.6, h φ 1 is a hypotrochoid up to a rotation around the point b 1 e iφ 1 attaining absolute values in the annulus A φ 1 with outer radius b 2 + b 3 and inner radius b 2 − b 3 around the point b 1 e iφ 1 (A φ 1 degenerates to a disc for b 2 = b 3 ). Since |b 2 | ≥ |b 3 |, it follows from Lemma 4.4 that h 0 intersects the real coordinate axis in both at least one point greater or equal than b 1 and at least one point less or equal than b 1 . More specific, let φ 2 (1), . . . , φ 2 (k) ∈ [0, 2π) denote the arguments such that µ j = g(0, φ 2 (j)) ∈ R with µ 1 ≥ · · · ≥ µ k . Analogously, we denote by φ 2 (1) ′ , . . . , φ ′ 2 (l) ∈ [0, 2π) the arguments such that ν j = g(π, φ ′ 2 (j)) ∈ R with ν 1 ≥ · · · ≥ ν l . Note that The key observation of the proof is the following: h φ 1 depends continuously on φ 1 . But this means that is a homotopy of hypotrochoid curves along the circle with radius b 1 . Since Since g(0, φ 2 ) is a real hypotrochoid, i.e., in particular, connected and symmetric along the real line, for every 1 ≤ j ≤ k − 1 there exists a closed connected subset γ j of the trajectory of the hypotrochoid g(0, φ 2 ) and its pointwise complex conjugate γ j both connecting µ j and µ j+1 . I.e., ρ j = γ j ∪ γ j forms a topological circle intersecting R exactly in µ j and µ j+1 and thus its projection on R covers [µ j+1 , µ j ]. Hence, k−1 j=1 ρ j projected on the real line covers [µ k , µ 1 ]. Now, we restrict the homotopy H of hypotrochoids to a particular circle ρ j and to moving φ 1 continuously from 0 to π, i.e., the induced homotopy is H j : ρ j × [0, 1] → C of the circle ρ j moved around the half-circle {b 1 e i·φ 1 : φ 1 ∈ [0, 2π)}. Two cases can occur during the homotopy H j : Either R intersects the circle ρ j transversally in two points during the whole homotopy, or there exists a point τ ∈ (0, 1) such that the circle and R intersect non-transversally at H j (ρ j , τ ).

Sums of Squares supported on a circuit
In this section we completely characterize the section Σ y n,2d . It is particularly interesting that this section depends heavily on the lattice point configuration in ∆, thereby, yielding a connection to the theory of lattice polytopes and toric geometry. By investigating this  connection in more detail, we will prove that the sections P y 2,2d and Σ y 2,2d almost always coincide and that P y n,2d and Σ y n,2d contain large sections, at which nonnegative polynomials are equal to sums of squares for n > 2, see Corollaries 5.10 and 5.12.
Surprisingly, the sums of squares condition is exactly the same as for the corresponding agiforms. For this, we briefly review the Gram matrix method for sums of squares polynomials. For d ∈ N let N n d = {α ∈ N n : α 1 + · · · + α n ≤ d} and p = r In this case, [B(β) · B(β ′ )] β,β ′ ∈N n d is a positive semidefinite matrix. Furthermore, we need the following well-known lemma, see [3].
Lemma 5.1. Let f ∈ Σ n,2d be a sum of squares and T ∈ GL n (R) be a matrix yielding a variable transformation x → T x. Then f (T x) also is a sum of squares. Now, we can characterize the sums of squares among nonnegative polynomials in P y ∆ . Theorem 5.2. Let f = λ 0 + n j=1 b j x α(j) + c · x y ∈ P y n,2d . Then f ∈ Σ y n,2d if and only if y ∈ ∆ * or c > 0 and y ∈ (2N) n .
Furthermore, if f ∈ Σ y n,2d , then f is a sum of binomial squares. Note again that for f ∈ P y ∆ the condition c > 0 and y ∈ (2N) n holds if and only if f is a sum of monomial squares such that the above theorem holds trivially.
Let now y ∈ ∆ * . We investigate two cases. First, let y / ∈ (2N) n . Then it suffices to prove the statement for c = ±Θ f by the following argument: Let f 1 = λ 0 + n j=1 b j x α(j) −c·x y ∈ P y n,2d and f 2 = λ 0 + n j=1 b j x α(j) + c · x y ∈ P y n,2d . Let c * be such that −c < c * < c and f 3 = λ 0 + n j=1 b j x α j + c * · x y ∈ P y n,2d . Then we have f 3 = λ 1 f 1 + λ 2 f 2 with λ 1 = c+c * 2c , λ 2 = c−c * 2c and λ 1 , λ 2 > 0, λ 1 + λ 2 = 1. By the same argument involving the variable transformation x j → −x j for some j ∈ {1, . . . , n} as before (proof of Theorem 3. If y ∈ (2N) n , then we use the same argument to prove that f is a sum of squares for c = −Θ f . For c > −Θ f , the polynomial f is obviously a sum of squares, since the inner monomial can be written as −Θ f x y plus the term (c + Θ f )x y , which is a square.
In [31,Theorem 4.4] it is shown that the agiforms in (5.1) are sums of binomial squares. Thus, for y ∈ ∆ * , the nonnegative polynomials f ∈ P y n,2d are also sums of binomial squares, since the binomial structure is preserved under the variable transformation T . Agiforms can be recovered by setting b j = λ j and, hence, Theorems 3.8 and 5.2 generalize results for agiforms in [31]. Furthermore, by setting α(j) = 2d · e j for 1 ≤ j ≤ n, we recover the dehomogenized version of what is called an elementary diagonal minus tail form in [9], and, again, Theorems 3.8 and 5.2 generalize one of the main results in [9] to arbitrary simplices.
We remark that in [31] an algorithm is given to compute such a sum of squares representation in the case of agiforms in Theorem 5.2, which can be generalized to arbitrary circuit polynomials. Furthermore, in [31] it is shown that every agiform in Σ y n,2d can be written as a sum of |L \∆| binomial squares. By using the variable transformation T in the proof of Theorem 5.2, we conclude that a general circuit polynomial f ∈ Σ y n,2d also can be written as a sum of |L \∆| = |L| − (n + 1) binomial squares.
Theorem 5.2 also comes with two immediate corollaries. Corollary 5.3. Let ∆ be an H-simplex and f ∈ P y ∆ . Then f ∈ P y n,2d if and only if f ∈ Σ y n,2d . Proof. Since ∆ is an H-simplex, it holds that ∆ * = (∆ ∩ Z n ) (see Section 2.3) and we always have y ∈ ∆ * .
The second corollary concerns sums of squares relaxations for minimizing polynomial functions. For this, note that the quantity f * sos = max{λ : f − λ ∈ Σ n,2d } is a lower bound for f * = min{f (x) : x ∈ R n }, see for example [21].
∆ . Then f * sos = f * if and only if y ∈ ∆ * . Proof. We have f * sos = f * if and only if f −f * ∈ Σ n,2d . However, subtracting the minimum of the polynomial f does not affect the question whether y ∈ ∆ * or not. Hence, if y ∈ ∆ * , this will also hold for the nonnegative polynomial f − f * and vice versa.
As an extension, we consider in the following the case of multiple support points, which are interior lattice points in the simplex ∆ = conv{0, α(1), . . . , α(n)}. Assume that all interior monomials come with a negative coefficient. Then we can write the polynomial as a sum of nonnegative circuit polynomials if and only if it is nonnegative. Furthermore, we get an equivalence between nonnegativity and sums of squares if the whole support is contained in ∆ * . In the following, let {λ where all E y(i) ∈ P y(i) ∆(i) are nonnegative with support sets ∆(i) ⊆ {0, α(1), . . . , α(n), y(i)}.
If furthermore I ⊆ ∆ * , then we have f ∈ P n,2d if and only if f ∈ Σ n,2d (5.2) if and only if f is a sum of binomial squares.
Again, we get an immediate corollary.
Corollary 5.6. Let f be as above with I ⊆ ∆ * . Then f * sos = f * . In order to prove Theorem 5.5, we need the following lemma.
be nonnegative with simplex Newton polytope New(f ) = ∆ = conv{0, α(1), . . . , α(n)} for some α(j) ∈ (2N) n . Furthermore, let I ⊆ (int(∆) ∩ N n ) and a i , b j > 0. Then f has a global minimizer v * ∈ R n >0 . Proof. Since all b j > 0 and α(j) ∈ (2N) n , clearly f has a global minimizer on R n . Assume that all global minimizers are not contained in R n ≥0 . We make a term by term inspection for a minimizer v in comparison with |v| = (|v 1 |, . . . , |v n |): For every vertex of ∆ we have b j v α(j) = b j |v α(j) |; for every interior point we have −a i |v| y(i) ≤ −a i v y(i) and hence f (v) ≥ f (|v|). This is a contradiction and therefore at least one global minimizer v * is contained in R n ≥0 . Assume that for at least one component v * j of v * it holds that v * j = 0. We define g = b 0 + n j=1 b j x α(j) − a i x y(i) for one y(i) ∈ I. By Proposition 3.3, g(e w ) has a unique global minimizer on R n and hence g has a unique global minimizer on R n >0 . But, by construction of f and g, we have f (x) < g(x) for all x ∈ R n >0 and f (x) = g(x) for x ∈ R n ≥0 \ R n >0 . Thus, v * j = 0 for all 1 ≤ j ≤ n.
Proof. (Proof of Theorem 5.5) Let f = |I| j=1 λ (j) 0 + n j=1 b j x α(j) − y(i)∈I a i x y(i) be nonnegative and, by Lemma 5.7, let v ∈ R n >0 be a global minimizer of f . First, we investigate the case α(j) = α j e j for some α j ∈ 2N * and e j denoting the j-th standard vector. For any 1 ≤ k ≤ n we have n be the coefficients of the unique convex combination of y(i) ∈ I and λ (i) = (λ Since for all i and all k it holds that n j=1 λ (i) k α(j) k = y(i) k and that all α(j) k = 0 unless j = k, we obtain with (5.3) that

By Proposition 3.4 and Theorem 3.8, we conclude that
is a nonnegative circuit polynomial and has its minimum value at v. We obtain Now, we consider the case of arbitrary α(j) ∈ (2N) n . Let v ∈ R n >0 be a global minimizer of f . By Corollary 3.2 (and Proposition 3.1) there exists a unique polynomial g satisfying f (e w ) = g(e T t w ) for all w ∈ R n (5.6) such that T ∈ GL n (R) and g has a support matrix where µ is the least common multiple of the denominators of all λ (i) j and 2 (since vertices of New(g) shall be in (2N) n ).
Since v ∈ R n >0 , we can define Log |v ′ | = T t Log |v|. By (5.5) and (5.6) it follows that v ′ is a global minimizer for g and thus we have for some nonnegative circuit polynomials E µλ (i) with global minimizer v ′ ∈ R n >0 . Since supp(E µλ (i) ) ⊆ supp(g) and New(E µλ (i) ) = New(g), we have, by Proposition 3.4, such that each E y(i) (e Log |v| ) is a nonnegative circuit polynomial with global minimizer v and support set {0, α(1), . . . , α(n), y(i)} satisfying f = |I| i=1 E y(i) .
If, additionally, every y(i) ∈ ∆ * (for example if ∆ is an H-simplex), then we know by Theorem 1.2 that all E y(i) (x) are sums of (binomial) squares and, hence, f is a sum of (binomial) squares.
Note that Theorem 5.5 generalizes [9, Theorem 2.7], where an analog statement is shown for the special case of diagonal minus tail forms f , which are given by α(j) = 2d for 1 ≤ j ≤ n.
We remark that the correct decomposition of the b j in Theorem 5.5 for the case of a general simplex Newton polytope is also given by (5.4), since due to these scalars remain invariant under the transformation T from and to the standard form.
Example 5.8. The polynomial f = 1 + 1 2 x 6 + 1 32 y 4 − 1 2 xy − 1 2 x 2 y is nonnegative and has a zero at v = (1, 2). By using the constructions in Theorem 5.5, we can decompose f as sum of two polynomials in P y n,2d with y ∈ {(1, 1), (2, 1)} and vanishing at v. More precisely, Since ∆ is an H-simplex, we have f ∈ Σ 2,6 . Using the algorithm in [31] and a suitable variable transformation (see proof of Theorem 5.2), we get the following representation for f as a sum of binomial squares:

A Sufficient
Condition for H -simplices. By Theorem 5.2, all nonnegative polynomials in P y ∆ supported on an H-simplex are sums of squares. Here, we provide a sufficient condition for a lattice simplex ∆ to be an H-simplex, meaning, that all lattice points in ∆ except the vertices are midpoints of two even distinct lattice points in ∆. In the following, we call a full dimensional lattice polytope P ⊂ R n k-normal, if every lattice point in kP is a sum of exactly k lattice points in P , i.e., For an introduction to toric ideals, see for example [36]. . By clearing denominators in the unique convex combination of u we get a relation For the corresponding toric ideal I B , this implies that Since I B is generated in degree two, we have the following representation: for some polynomials f m,n . Matching monomials, it follows that there exists m such that i.e., 2u is a convex combination of two even lattice points 2v and 2v ′ .
Corollary 5.10. Let ∆ ⊂ R 2 be a lattice simplex as in Theorem 5.9 such that 1 2 ∆ has at least four boundary lattice points. Then ∆ is an H-simplex.
Proof. Since every 2-polytope is normal, we only need to prove that the corresponding toric ideal is generated in degree two. But this is [19,Theorem 2.10].
Hence, in R 2 , almost every simplex ∆ corresponding to P y ∆ is an H-simplex, which is a fact that was announced in [31] without proof. This implies that the sections P y 2,2d and Σ y 2,2d almost always coincide. Example 5.11. We demonstrate Theorem 5.9 by two interesting examples.
In higher dimensions things get more involved both in checking the conditions in Theorem 5.9 and in determining the maximal∆-mediated set ∆ * . Note that ∆ * can lie strictly between A(∆) and ∆ ∩ Z n , which correspond to M-simplices and H-simplices. In [31] an algorithm for the computation of ∆ * is given. One expects the existence of better algorithms, but, to our best knowledge, no more efficient algorithm is known. On the other hand, checking normality of polytopes and quadratic generation of toric ideals is an active area of research. It is an open problem to decide, whether every smooth lattice polytope is normal and the corresponding toric ideal is generated by quadrics, see [15,36]. However, for an arbitrary lattice polytope P the multiples kP are normal for k ≥ dim P − 1 and their toric ideals are generated by quadrics for k ≥ dim P [5]. In light of these results, we can conclude another interesting corollary from Theorem 5.9.
Corollary 5.12. Let ∆ ⊂ R n be a lattice simplex as in Theorem 5.9 such that 1 2 ∆ = M∆ ′ for a lattice simplex ∆ ′ ⊂ R n and M ≥ n. Then ∆ is an H-simplex.
Proof. The result follows from the previously quoted results together with Theorem 5.9.
Note that Corollaries 5.10 and 5.12 yield large sections at which nonnegative polynomials and sums of squares coincide.

Convex Polynomials and Forms Supported on Circuits
In this section, we investigate convex polynomials and forms (i.e., homogeneous polynomials) supported on a circuit. Recently, there is much interest in understanding the convex cone of convex polynomials/forms. Since deciding convexity of polynomials is NPhard in general [1], but very important in different areas in mathematics, such as convex optimization, the investigation of properties of the cones of convex polynomials and forms is a genuine problem.
Unlike the property of nonnegativity and sums of squares, convexity of polynomials is not preserved under homogenization. Therefore, we need to distinguish between convex polynomials and convex forms. The relationship between convexity on the one side and nonnegativity and sums of squares on the other side arises when considering homogeneous polynomials, since every convex form is nonnegative. However, the relation between convex forms and sums of squares is not well understood except for the fact that their corresponding cones are not contained in each other. The problem to find a convex form that is not a sum of squares is still open. For an overview and proofs of the previous facts see [3,32]. Here we investigate convexity of polynomials and forms in the class P y ∆ . We start with the univariate (nonhomogeneous) case. Proposition 6.2. Let f = 1 + ax y + bx 2d ∈ P y ∆ and b > 0. Then f is convex exactly in the following cases.
Proof. Let f = 1 + ax y + bx 2d . Note that the degree is necessarily even and b > 0. f is convex if and only if D 2 (f ) ≥ 0 where D 2 (f ) = ay(y − 1)x y−2 + 2db(2d − 1)x 2d−2 . For y = 1 the polynomial D 2 (f ) is a square and hence f is convex. Now, consider the case y > 1. First, suppose that a < 0. Then D 2 (f ) is always indefinite, since the monomial x y−2 in D 2 (f ) corresponds to a vertex of the corresponding Newton polytope of D 2 (f ) and has a negative coefficient. Otherwise, if a ≥ 0 and y = 2l for l ∈ N, then D 2 (f ) ≥ 0 and f is convex. If y = 2l + 1, then x y−2 has an odd power and hence D 2 (f ) is indefinite, implying that f is not convex.
The homogeneous version is much more difficult than the affine version. We just prove the following claims instead of giving a full characterization. Proposition 6.3. Let f = z 2d + ax y z 2d−y + bx 2d ∈ P y ∆ be a form and b > 0. Then the following hold.
Evaluating this partial derivative at z = 1, in order to be nonnegative, it is obvious that y must be even and a ≥ 0, proving the first claim. For the second claim, we investigate the principal minors of H f . We have that . This yields y = 1 or a ≥ 0 and y = 2l. From . Hence, for y = 2l and 0 ≤ a ≤ (y−1)(2d−y−1) the form f is convex.
Note that for y = 1 the form f = z 2d + ax y z 2d−y + bx 2d ∈ P y ∆ is never convex, whereas, by Proposition 6.2, the dehomogenized polynomial is always convex. As a sharp contrast, we prove the surprising result that for n ≥ 2 there are no convex polynomials in the class P y ∆ , implying that there are no convex forms in P y ∆ for n ≥ 3. Theorem 6.4. Let n ≥ 2 and f ∈ P y ∆ . Then f is not convex. Proof. Let . · x yn n with A j > 0 for 1 ≤ j ≤ n and B ∈ R * . We will prove that the principal minor [1,2] × [1,2] (deleting all rows and columns except the first and second one) of the Hessian of f is indefinite, implying that the Hessian of f is not positive semidefinite and, hence, the polynomial f is not convex. We have We claim that there is a point x ∈ R n at which this minor is negative.
(α(j) 1 + y 1 − 2, α(j) 2 + y 2 − 2, α(j) 3 + y 3 , . . . , α(j) n + y n ) for 1 ≤ j ≤ n, (4) (2y 1 − 2, 2y 2 − 2, 2y 3 , . . . , 2y n ). We claim that the point (2y 1 −2, 2y 2 −2, 2y 3 , . . . , 2y n ) is always a vertex in the convex hull of the points (1)-(4), i.e., in the Newton polytope of the investigated minor. The points in (2) are obviously convex combinations from appropriate points in (1) and the points in (3) are convex combinations from points in (1) and (4). Hence, it remains to show that (4) is not a convex combination of the points in (1). Therefore, denote the points in (1) by P j and the point in (4)  But this means that y lies on the boundary of ∆, the Newton polytope of f . This is a contradiction, since f ∈ P y ∆ , i.e., y ∈ int(∆). Hence, (4) is a vertex of the Newton polytope of the investigated minor. Extracting the coefficient of its corresponding monomial in the minor, we get that this coefficient equals −B 2 y 1 y 2 (y 1 +y 2 −1) < 0. Therefore, the Newton polytope of the minor of the Hessian of f has a vertex coming with a negative coefficient and, hence, it is indefinite, proving the claim.
Note that this already implies that there is also no convex form in P y ∆ whenever n ≥ 3, since non-convexity is preserved under homogenization. Since it is mostly unclear which structures prevent polynomials from being convex, Theorem 6.4 is an indication that sparsity is among these structures.

Sums of Nonnegative Circuits
Motivated by results in previous sections, we recall Definition 1.3 from the introduction, where we introduced sums of nonnegative circuit polynomials (SONC's), a new family of nonnegativity certificates. Definition 7.1. We define the set of sums of nonnegative circuit polynomials (SONC) as for some even lattice simplices ∆ i ⊂ R n .
Remember that membership in P y n,2d can easily be checked and is completely characterized by the circuit numbers Θ g i (Theorem 3.8). Obviously, for α, β ∈ R >0 and f, g ∈ C n,2d , it holds that αf + βg ∈ C n,2d , hence, C n,2d is a convex cone. Then we have the following relations.
Proposition 7.2. The following relationships hold between the corresponding cones.
Hence, the convex cone C n,2d serves as a nonnegativity certificate, which, by Proposition 7.2, is independent from sums of squares certificates. We give two further remarks about the Proposition 7.2: (1) As stated in the proof (n, 4) is the case, which is not covered in Part (3). We believe that Σ n,4 ⊂ C n,4 for all n but we do not have an example. (2) Let C n,2d be the subset of C n,2d containing all polynomials with a full dimensional Newton polytope. It is not obvious for which cases next to (n, 2d) ∈ {(1, 2d), (n, 2), (2, 4)} it holds that C n,2d ⊆ Σ n,2d . However, C n,2d ⊆ Σ n,2d if we require d ≥ n + 1 as we show in the following example.
Of course, a priori it is completely unclear for which type of nonnegative polynomials a SONC decomposition exists and how big the gap between C n,2d and P n,2d is. Furthermore, it is not obvious how to compute such a decomposition, if it exists. We discuss this question in a follow up article [17]. In this article we show in particular that for simplex Newton polytopes (with arbitrary support) such a decomposition exists if and only if a particular geometric optimization problem is feasible, which can be checked very efficiently. This generalizes similar results by Ghasemi and Marshall [12,13]. Here we deduce as a fruitful first step the following corollary from Theorem 5.5.
Proof. Every monomial square is a strictly positive term as well as a 0-simplex circuit polynomial. Thus, we can ignore these terms. If a particular vector v ∈ (R * ) n with the desired properties exists, then Theorem 5.5 immediately yields a SONC decomposition after a variable transformation x j → −x j for all j with v j < 0.

Extension to Arbitrary Polytopes and Counterexamples
In Section 5 we proved for f ∈ P y ∆ that f ∈ Σ y n,2d if and only if y ∈ ∆ * or f is a sum of monomial squares. One might wonder whether this equivalence also holds for arbitrary polytopes. More precisely, let Q ⊂ R n be an arbitrary lattice polytope and denote by AP y Q the set of all polynomials of the form α∈vert(Q) b α x α + cx y that are supported on the vertices vert(Q) of Q and an additional interior lattice point y ∈ int(Q). As a generalization of our previous notation, we call f ∈ AP y Q an agiform if α∈vert(Q) b α α = y and α∈vert(Q) b α = 1 as well as b α > 0 and c = −1.
In [31,Section 10], it is asked, whether the lattice point criterion y ∈ Q * is again an equivalent condition for a polynomial in AP y Q to be a sum of squares. And, if not, how sums of squares can be characterized in this case. Here, we provide a solution to this question (Theorem 8.2). Let P y Q respectively Σ y Q denote the set of nonnegative respectively sums of squares polynomials in AP y Q . As for a simplex ∆, for an arbitrary lattice polytope Q, we use the same definition of an M-polytope respectively an H-polytope.
The implication f ∈ Σ y Q ⇒ y ∈ Q * does always hold. For agiforms, this is proven already in [31]. The proof in the case of arbitrary coefficients follows exactly the same line as the proof of Theorem 5.2.
It is easy to check that Q is an H-polytope (indeed, it can actually be proven that Theorem 5.9 is true for arbitrary polytopes not just for simplices). Since Q is not a simplex, there are infinitely many convex combinations of y: y = λ 0 v 1 + λ 1 v 1 + λ 2 v 2 + λ 3 v 3 such that 3 i=0 λ i = 1 and λ i ≥ 0.
The set of convex combinations of y is given by The corresponding agiform f (Q, λ, y) is then given by For λ 3 = 2 5 , the nonnegative polynomial f = 3 10 + 1 10 x 4 + 1 5 x 4 y 2 + 2 5 x 2 y 4 − x 2 y 2 can easily be checked to be not a sum of squares although y ∈ Q * via the corresponding Gram matrix.
Actually, one can prove that the polynomial f (Q, λ, y) in the above proof is a sum of squares if and only if λ 3 = 1 2 . In [31], the author suspects that the condition y ∈ Q * is not sufficient by looking at similar examples. However, in all of these examples, the constructed polynomials that are nonnegative but not a sum of squares are not supported on the vertices of Q and an additional interior lattice point y ∈ int(Q). We conclude that in the non-simplex case the problem of deciding the sums of squares property depends on the coefficients of the polynomials, a sharp contrast to the simplex case. However, motivated by a question in [31] for agiforms, we are interested in the following sets: Let C(y) denote the set of convex combinations of the interior lattice point y ∈ int(Q), i.e., C(y) = λ = (λ 0 , . . . , λ s ) : where v i are the s vertices of Q. Note that C(y) is a polytope. Fixing f and y, we define SOS(f, y) = {λ ∈ C(y) : f (Q, λ, y) is a sum of squares} where Q = New(f ). We have already seen in the proof of Proposition 8.1 that the structure of SOS(f, y) is unclear and highly depends on the convex combinations of y. It is formulated as an open question in [31], whether one can say something about SOS(f, y) for fixed f and y. For this, let Q = Q (i) 1 ∪ · · · ∪ Q (i) r(i) be a triangulation of Q for 1 ≤ i ≤ t, where t is the number of triangulations of Q without using new vertices. We are interested in those simplices Q (i) j that contain the point y ∈ int(Q) and their maximal mediated sets (Q (i) j ) * . Recall that for every lattice simplex ∆ with vertex set∆ we denote ∆ * as the maximal∆-mediated set (see Section 2.3).
Theorem 8.2. Let Q ⊂ R n be a lattice n-polytope, y ∈ int(Q) ∩ N n and f ∈ AP y Q be an agiform. Then SOS(f, y) = C(y), i.e., every agiform is a sum of squares, if and only if y ∈ Q (i) j implies y ∈ (Q (i) j ) * for every 1 ≤ i ≤ t and 1 ≤ j ≤ r(i).
Proof. Assume y ∈ Q (i) j ⇒ y ∈ (Q (i) j ) * for every 1 ≤ i ≤ t and 1 ≤ j ≤ r(i). Let λ ∈ C(y) with f (Q, λ, y) being the corresponding agiform. By [31,Theorem 7.1], every agiform can be written as a convex combination of simplicial agiforms. In fact, following the proof in [31,Theorem 7.1], it can be verified that the vertices of the corresponding simplicial agiforms form a subset of the vertices of Q, since the set C(y) of convex combinations of y is a polytope with vertices being a subset of vert(Q). Hence, these agiforms come from triangulating the polytope Q into simplices without using new vertices. Since y ∈ Q (i) j ⇒ y ∈ (Q (i) j ) * for every i, j, by Theorem 2.4, the corresponding simplicial agiforms are always sums of squares and since f (Q, λ, y) is a sum of them, the claim follows.
For the reverse direction, assume y ∈ Q be the corresponding agiforms. Note that the coefficients λ i depend on d parameters µ 1 , . . . , µ d , since dim C(y) = d. By assumption, there exist a 1 , . . . , a d ∈ R >0 such that the corresponding agiform f (Q, λ, y) |(µ 1 ,...,µ d )=(a 1 ,...,a d ) = g is a simplicial agiform with respect to the simplex Q (i) j . Since y ∈ Q (i) j but y / ∈ (Q (i) y,k ) * , the agiform g is not a sum of squares. By continuity, we can construct a sequence (µ 1 , . . . , µ d ) converging against (a 1 , . . . , a d ) with the properties that f (Q, λ, y) |(µ 1 ,...,µ d )=(a 1 +ε,...,a d +ε) is an agiform for some ε > 0 with its support equal to {v 1 , . . . , v m , y} and not being a sum of squares, since, otherwise, if every sequence member is a sum of squares, this will also hold for the limit agiform g corresponding to (a 1 , . . . , a d ) since the cone of sums of squares is closed. Hence, SOS(f, y) = C(y). Since Q has four vertices, C(y) for y ∈ (int(Q) ∩ N n ) has a free parameter λ 3 (see proof of Proposition 8.1). In the following table, for all y ∈ (int(Q) ∩ N n ), we provide the range of the free parameter λ 3 yielding valid convex combinations for y as well as the set SOS(f, y). 7, the case of polynomials with simplex Newton polytopes is solved in [17] via geometric programming generalizing earlier work by Ghasemi in Marshall [12,13].
From the viewpoint of amoeba theory one evident conjecture is that Theorem 4.2 can be generalized to arbitrary complex polynomials supported on a circuit. Taking into account the corresponding literature, in particular [27,37], an answer to this conjecture can be considered as the final piece missing in order to completely characterize amoebas supported on a circuit.
In our opinion, the most interesting question is whether similar approaches can be generalized to more general (sparse) polynomials and, in accordance, how much deeper the observed connection between the a priori very distinct mathematical topics "amoebas" and "nonnegativity of real polynomials" is? We believe that exploiting methods from amoeba theory might eventually yield fundamental progress in understanding nonnegativity of real polynomials.