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In Davis and Papp (2022) , the authors introduced the concept of dual certificates of (weighted) sum-of-squares polynomials, which are vectors from the dual cone of weighted sums of squares (WSOS) polynomials that can be interpreted as nonnegativity certificates. This initial theoretical work showed that for every polynomial in the interior of a WSOS cone, there exists a rational dual certificate proving that the polynomial is WSOS. In this article, we analyze the complexity of rational dual certificates of WSOS polynomials by bounding the bit sizes of integer dual certificates as a function of parameters such as the degree and the number of variables of the polynomials, or their distance from the boundary of the cone. After providing a general bound, we explore several special cases, such as univariate polynomials nonnegative over the real line or a bounded interval , represented in different commonly used bases. We also provide an algorithm which runs in rational arithmetic and computes a rational certificate with boundable bit size for a WSOS lower bound of the input polynomial. Introduction It is well known that nonnegative polynomials with rational coefficients in the interior of sum-of-squares cones are sums of rational squares ; that is, they have sum-of-squares decompositions that are expressed entirely in terms of rational coefficients and can be verified using rational arithmetic Powers (2011). The complexity of these rational certificates of nonnegativity can be measured by the bit size of the largest magnitude coefficient in the decomposition; bounding the complexity of the “simplest” certificate and establishing its dependence on relevant parameters such as the degree, the number of variables, or the polynomial's distance from the boundary of the cone are major open questions. The corresponding algorithmic question is how efficiently these decompositions can be computed in rational arithmetic. Surprisingly, polynomial-time algorithms are difficult to design, and tight complexity bounds of known sum-of-squares decomposition algorithms are hard to come by even in the univariate case Magron et al. (2019). The recent paper by Magron and Safey El Din (2021a) gives an in-depth review of the state-of-the-art on the complexity of deciding and certifying the nonnegativity or positivity of polynomials over basic semialgebraic sets using SOS certificates; we only recall a few highlights. The paper Magron et al. (2019) focuses on univariate nonnegative polynomials over the real line. The most efficient algorithm they analyze returns SOS certificates of bit size O ( d 4 + d 3 τ ) , wherein d is the degree of the polynomial and τ is the bit size of the largest magnitude coefficient; a slight improvement over the results in Boudaoud et al. (2008), which consider (Pólya-type) WSOS certificates of positivity for univariate polynomials over [ − 1 , 1 ] . These algorithms cannot be generalized to the multivariate case. The multivariate case is considered first in Magron and Safey El Din (2021b), and substantially corrected in the report Magron and Safey El Din (2021a), by analyzing the bit complexity of a hybrid numerical-symbolic algorithm that recovers exact rational WSOS decompositions from an approximate (numerical) WSOS decomposition of a suitably perturbed polynomial. The main result in the corrected manuscript is that for n -variate SOS polynomials of degree d , the coefficients in the SOS certificates have bit sizes of order O ( τ d d O ( n ) ) . In this paper, we study these questions in the context of dual certificates . Dual certificates were introduced in Davis and Papp (2022) by the authors, motivated by (and building on) the duality theory of convex conic optimization, which has seen a number of recent applications in real algebraic geometry Katthän et al. (2021); Papp (2023). They are rational vectors from the dual cone of WSOS polynomials that by definition can be represented as a vector with far fewer components than a conventional WSOS decomposition: their dimension is independent of the number of weights, and they avoid the explicit representation of the large positive semidefinite Gram matrices that characterize conventional SOS decompositions. In Davis and Papp (2022), it was established that polynomials in the interior of a WSOS cone have rational dual certificates, also providing new elementary proofs of Powers's theorems from Powers (2011). Following up on this work, we now study the bit sizes of the components of dual certificates, as well as exact-arithmetic algorithms for the computation of rational dual certificates. In the first part of the paper, we show that dual certificates can be rounded (trivially, componentwise) to “nearby” rational dual certificates with computable, “small” denominators. This follows from a quantitative version of a property of dual certificates that (in contrast to conventional WSOS decompositions) every WSOS polynomial has a full-dimensional cone of dual certificates. In turn, these rational certificates can be converted to integer dual certificates with boundable bit size. In Section 2, we establish our general results, which are applicable to any WSOS cone certifying nonnegativity over arbitrary basic, closed semialgebraic sets, including unbounded ones. We then provide refinements for the most frequently studied and applied special cases, including univariate polynomials over the real line and over bounded intervals in Section 3. E.g., for univariate polynomials over the real line, we show that every positive polynomial with integer coefficients of bit size at most τ (in the monomial basis) has an integer dual certificate whose components are of bit size O ( d τ + d log ⁡ ( d ) ) —an improvement from the aforementioned result of Magron et al. (2019) and from the n = 1 special case of the bounds obtained in Magron and Safey El Din (2021a). In the second part of the paper (Section 4), we provide an algorithm that takes a polynomial with rational coefficients as its input, and computes a sequence of rational lower bounds converging to the optimal sum-of-squares lower bound, along with a corresponding sequence of rational dual certificates certifying these bounds. The algorithm is based on the one proposed in Davis and Papp (2022), which is an almost entirely numerical hybrid method. Although the method in Davis and Papp (2022) is capable of computing rational lower bounds and dual certificates via floating point computations, it is limited by the precision of the floating point arithmetic, and the bit sizes of the computed certificates cannot be bounded. The new algorithm proposed in this paper, Algorithm 1, runs entirely in infinite precision (rational) arithmetic. We show that all intermediate computations can be carefully rounded to nearby rational vectors with small denominators in each step, while still maintaining the property that the algorithm converges q -linearly to the optimal weighted sums-of-squares lower bound. Here, we cover notation and background that we will use throughout the rest of this paper. Recall that a convex set K ⊆ R n is called a convex cone if for every x ∈ K and λ ≥ 0 scalar, the vector λ x also belongs to K . A convex cone is proper if it is closed, full-dimensional (meaning span ( K ) = R n ), and pointed (that is, it does not contain a line). We shall denote the interior of a proper cone K by K ∘ and the boundary of a proper cone K by bd ( K ) . The dual of a convex cone K ⊆ R n is the convex cone K ⁎ defined as K ⁎ = { y ∈ R n | ∀ x ∈ K : x T y ≥ 0 } . Let V n , 2 d denote the cone of n -variate polynomials of degree 2 d . We say that a polynomial p ∈ V n , 2 d is sum-of-squares (SOS) if there exist polynomials q 1 , … , q k ∈ V n , d such that p = ∑ i = 1 k q i 2 . Define Σ n , 2 d to be the cone of n -variate SOS polynomials of degree 2 d . The cone Σ n , 2 d ⊂ V n , 2 d ≡ R ( n + 2 d n ) is a proper cone for every n and d . Throughout, we will identify polynomials with their coefficients vectors (typeset bold) in a basis that is clear from the context (but not necessarily in the monomial basis), e.g., t for the polynomial t ( ⋅ ) and 1 for the constant one polynomial. More generally, let w = ( w 1 , … , w m ) be some given nonzero polynomials and let d = ( d 1 , … , d m ) be a nonnegative integer vector. We denote by V n , 2 d w the space of polynomials p for which there exist r 1 ∈ V n , 2 d 1 , … , r m ∈ V n , 2 d m such that p = ∑ i = 1 m w i r i . A polynomial p ∈ V n , 2 d w is said to be weighted sum-of-squares (WSOS) if there exist σ 1 ∈ Σ n , 2 d 1 , … , σ m ∈ Σ n , 2 d m such that p = ∑ i = 1 m w i σ i . It is customary to assume that w 1 = 1 , that is, the ordinary “unweighted” sum-of-squares polynomials are also included in the WSOS cones. Let Σ n , 2 d w denote the set of WSOS polynomials in V n , 2 d w . This is nearly identical to the notion of the truncated quadratic module , except that the degree of each SOS polynomial is independently selected, rather than by “truncating” to a desired total degree. In this manner, Σ n , 2 d w is automatically a full-dimensional convex cone in the ambient space V n , 2 d w by definition. Additionally, under mild conditions, the cone Σ n , 2 d w is closed and pointed; for example, it is sufficient that the set S w = def { x ∈ R n | w i ( x ) ≥ 0 , i = 1 , … , m } is a unisolvent point set for the space V n , 2 d w (Papp and Yıldız, 2019, Prop. 6.1). (A set of points S ⊆ R n is unisolvent for a space of polynomials V if every polynomial in V is uniquely determined by its function values at S .) In particular, this implies that both Σ n , 2 d w and its dual cone have non-empty interiors, a crucial assumption throughout the paper. We will denote the set of n × n real symmetric matrices by S n , and the cone of positive semidefinite n × n real symmetric matrices by S + n . When the dimension is clear from the context, we use the common shorthands A ≽ 0 to denote that the matrix A is positive semidefinite and A ≻ 0 to denote that the matrix A is positive definite. It is well-known and easily seen that a polynomial s belongs to Σ n , 2 d if and only if s ( ⋅ ) = v n , d ( ⋅ ) T S v n , d ( ⋅ ) , wherein v n , d denotes the vector of n -variate monomials up to degree d , and S ∈ S + L with L = ( n + d d ) is the Gram matrix of the SOS polynomial s . This functional equality can be expressed coefficient-by-coefficient, identifying the polynomials on both sides of the equation with their coefficient vectors in a fixed basis. For example, if n = 1 and both polynomials are represented in the monomial basis, we obtain the classic result that s ( t ) = ∑ i = 0 2 d s i t i is SOS if and only if there exists a matrix ( S j k ) j , k = 0 , … , d such that s i = ∑ ( j , k ) : i = j + k S j k for each i . More generally, every SOS cone Σ n , 2 d is a linear image of the cone S + L , and if we fix a basis for V n , 2 d and a basis for V n , d , there is an explicitly computable, surjective, linear map Λ ⁎ from Gram matrices (positive semidefinite matrices) to coefficient vectors of SOS polynomials. From the dual perspective, it is also well-known (and is equivalent to the above statements) that the dual cone Σ n , 2 d ⁎ is a linear pre-image of S + L . More precisely, there exists an injective linear map Λ : Σ n , 2 d ⁎ → S + L . In the context of algebraic geometry and moment theory, Λ ( y ) is called the truncated moment matrix of the (pseudo-)moment vector y , and the map Λ ⁎ in the representation of the SOS cone is simply the adjoint of Λ. E.g., in the univariate example above, Λ ( y ) is the Hankel matrix of the vector y . Although everything in the previous paragraph generalizes from SOS cones to the WSOS case, the conventional notation and terminology involving moment and localizing matrices is rather cumbersome, and is largely unnecessary for this paper. To follow the rest of the paper, it is sufficient to keep in mind that regardless of the number of variables n , the degree vector d , the choice of weights w , and the polynomial bases used to represent the polynomials of various degrees, the WSOS cone Σ n , 2 d w is a linear image of the cone of positive semidefinite matrices of appropriate size under some surjective linear map Λ ⁎ , and similarly, its dual ( Σ n , 2 d w ) ⁎ is a linear pre-image of the same cone, under the adjoint map Λ. The following Proposition makes these statements precise. Proposition 1.1 Nesterov, 2000, Thm. 17.6 Fix an ordered basis q = ( q 1 , … , q U ) of V n , 2 d w and an ordered basis p i = ( p i , 1 , … , p i , L i ) of V n , d i for each i = 1 , … , m . Let Λ i : V n , 2 d w ( ≡ R U ) → S L i be the unique (injective) linear map satisfying Λ i ( q ) = w i p i p i T , and let Λ i ⁎ denote its adjoint. Then s ∈ Σ n , 2 d w if and only if there exist matrices S 1 ≽ 0 , … , S m ≽ 0 satisfying s = ∑ i = 1 m Λ i ⁎ ( S i ) . Additionally, the dual cone of Σ n , 2 d w admits the characterization ( Σ n , 2 d w ) ⁎ = { x ∈ V n , 2 d w ( ≡ R U ) | Λ i ( x ) ≽ 0 ∀ i = 1 , … , m } . To see why Λ i exists and is unique, consider that each entry of the matrix of functions w i p i p i T is a polynomial of the form w i p i , j p i , k , which by definition belongs to the space V n , 2 d w , and so it can be written uniquely as a linear combination of our chosen basis polynomials { q 1 , … , q U } of this space. Thus, for any vector v ∈ R U , the ( j , k ) -th entry of the matrix Λ i ( v ) is the same linear combination of the components of v which would yield w i p i , j p i , k if it were applied to the basis polynomials q . The interested reader will find a number of examples of WSOS cones Σ and the Λ operators representing them in different bases in (Davis and Papp, 2022, Example 1), using the same notation as in this paper. We briefly recall only one of them: Example 1.2 Consider univariate polynomials of degree 4, nonnegative on [ − 1 , 1 ] . These polynomials can be written as σ 1 ( t ) + ( 1 − t 2 ) σ 2 ( t ) , where σ 1 ∈ Σ 1 , 4 and σ 2 ∈ Σ 1 , 2 ; that is, they are WSOS with the weights w 1 ( t ) = 1 and w 2 ( t ) = 1 − t 2 and degree vector d = ( 2 , 1 ) . Representing all monomials in the monomial basis, it is well-known that x = ( x 0 , … , x 4 ) ∈ ( Σ n , 2 d w ) ⁎ if and only if Λ 1 ( x ) : = ( x 0 x 1 x 2 x 1 x 2 x 3 x 2 x 3 x 4 ) ≽ 0 and Λ 2 ( x ) : = ( x 0 − x 2 x 1 − x 3 x 1 − x 3 x 2 − x 4 ) ≽ 0 . The matrix Λ 1 ( x ) is the moment matrix, while Λ 2 ( x ) is a localizing matrix for this particular domain Laurent (2009). In the notation of Proposition 1.1, we have U = 5 , ( L 1 , L 2 ) = ( 3 , 2 ) , and Λ i matrices were obtained by collecting the monomial terms in the matrices ( 1 t t 2 ) ( 1 t t 2 ) = ( 1 t t 2 t t 2 t 3 t 2 t 3 t 4 ) and ( 1 − t 2 ) ( 1 t ) ( 1 t ) = ( 1 − t 2 t − t 3 t − t 3 t 2 − t 4 ) . To further lighten the notation, throughout the paper, we will assume that the weight polynomials w = ( w 1 , … , w m ) and the degrees d = ( d 1 , … , d m ) are fixed. We will denote the cone Σ n , 2 d w by Σ and the space of polynomials V n , 2 d w by V . We will usually identify the spaces V and V ⁎ with R U ( U = dim ⁡ ( V ) ), equipped with the standard inner product 〈 x , y 〉 = x T y and the induced Euclidean norm ‖ ⋅ ‖ . For (real) square matrices, the inner product 〈 ⋅ , ⋅ 〉 denotes the Frobenius inner product. Additionally, we use the shorthand Λ to denote the R U → S L 1 ⊕ ⋯ ⊕ S L m linear map Λ 1 ( ⋅ ) ⊕ ⋯ ⊕ Λ m ( ⋅ ) from Proposition 1.1. With this notation, the condition (2) can be written as s = Λ ⁎ ( S ) for some positive semidefinite (block diagonal) matrix S ∈ S L 1 ⊕ ⋯ ⊕ S L m . Similarly, Eq. (3) simplifies to Σ ⁎ = { x ∈ R U | Λ ( x ) ≽ 0 } . The interior of this cone is simply ( Σ ⁎ ) ∘ = { x ∈ R U | Λ ( x ) ≻ 0 } . The theory of dual certificates builds heavily on results from the theory of barrier functions in convex optimization. Here, we introduce relevant notation, and we give a brief overview of the parts of this theory that will be needed throughout the rest of the paper. Let Λ : R U → S L be the unique linear mapping specified in Proposition 1.1 above, and let Λ ⁎ denote its adjoint. Central to our theory is the barrier function f : ( Σ ⁎ ) ∘ → R defined by f ( x ) = def − ln ⁡ ( det ⁡ ( Λ ( x ) ) . Note that by Eq. (4), f is indeed defined on its domain. The function f is twice continuously differentiable; we denote by g ( x ) its gradient at x and by H ( x ) its Hessian at x . Since f is strictly convex on its domain, H ( x ) ≻ 0 for all x ∈ ( Σ ⁎ ) ∘ (Boyd and Vandenberghe, 2004, Sec. 3.1.5 and 3.2.2). Consequently, we can also associate with each x ∈ ( Σ ⁎ ) ∘ the local inner product 〈 ⋅ , ⋅ 〉 x : V ⁎ × V ⁎ → R defined as 〈 y , z 〉 x = def y T H ( x ) z and the local norm ‖ ⋅ ‖ x induced by this local inner product. Thus, ‖ y ‖ x = ‖ H ( x ) 1 / 2 y ‖ . We define the local (open) ball centered at x with radius r by B x ( x , r ) = def { y ∈ V ⁎ | ‖ y − x ‖ x < r } . Analogously, we define the dual local inner product 〈 ⋅ , ⋅ 〉 x ⁎ : V × V → R by 〈 s , t 〉 x ⁎ = def s T H ( x ) − 1 t . The induced dual local norm ‖ ⋅ ‖ x ⁎ satisfies the identity ‖ t ‖ x ⁎ = ‖ H ( x ) − 1 / 2 t ‖ . Throughout, we will invoke several useful results concerning these norms and the barrier function f in (5); these are enumerated in Lemma A.1, in the Appendix of this paper. Geometrically, the key observation is that the Hessian of this barrier function, through the associated local and dual local norms, provides computable ellipsoidal neighborhoods around each point in Σ ∘ and ( Σ ⁎ ) ∘ that are contained in these cones, yielding “safe” bounds to round vectors in any direction without leaving the cone. As mentioned earlier in the introduction, our primary goal is to show the existence of a dual certificate with boundable bit size for a given WSOS polynomial. Here, we review necessary definitions and properties of dual certificates. For more extensive theory of dual certificates, see Davis and Papp (2022). Definition 1.3 Let s ∈ Σ , and denote the Hessian of the barrier function f of Σ ⁎ defined in (5) by H . We say that the vector x ∈ ( Σ ⁎ ) ∘ is a dual certificate of s , or simply that x certifies s , if H ( x ) − 1 s ∈ Σ ⁎ . We denote by C ( s ) = def { x ∈ ( Σ ⁎ ) ∘ | H ( x ) − 1 s ∈ Σ ⁎ } the set of dual certificates of s . Conversely, for every x ∈ ( Σ ⁎ ) ∘ , we denote by P ( x ) = def { s ∈ Σ | H ( x ) − 1 s ∈ Σ ⁎ } the set of polynomials certified by the dual vector x . This definition is motivated by the following theorem from Davis and Papp (2022), reproduced below for completeness. In words, the theorem provides an explicit closed form formula for efficiently computing a WSOS certificate for any polynomial from its coefficient vector s and any dual certificate x : Theorem 1.4 Davis and Papp, 2022, Thm. 2.2 Let s ∈ ( Σ ⁎ ) ∘ be arbitrary. Then the matrix S = S ( x , s ) defined by S ( x , s ) = def Λ ( x ) − 1 Λ ( H ( x ) − 1 s ) Λ ( x ) − 1 satisfies Λ ⁎ ( S ) = s . Moreover, x is a dual certificate for s ∈ Σ if and only if S ≽ 0 , which in turn is equivalent to H ( x ) − 1 s ∈ Σ ⁎ . Note that as long as Λ maps rational vectors to rational matrices (which is the case, for instance, when polynomials are represented in commonly used bases such as the standard monomial basis or the Chebyshev basis), then S is a rational matrix for every rational coefficient vector s . It is immediate from Definition 1.3 that if x is a dual certificate of the polynomial s , then every positive multiple of x is also a dual certificate for every positive multiple of s . Crucially, the same is true for small perturbations of x and s ; see Proposition 1.6 below. From Lemma A.1 (claim 5) in the Appendix, we know that for every s ∈ Σ ∘ there exists a unique x ∈ ( Σ ⁎ ) ∘ satisfying s = − g ( x ) . This vector is a dual certificate of s , since H ( x ) − 1 s = − H ( x ) − 1 g ( x ) = (A.6) x ∈ ( Σ ⁎ ) ∘ . Thus, every polynomial in the interior of the WSOS cone Σ has a dual certificate. Definition 1.5 When − g ( x ) = s ( ∈ Σ ∘ ) , we say that x is the gradient certificate of s . Simple calculus reveals the closed-form formula for the negative gradient: − g ( x ) = Λ ⁎ ( Λ ( x ) − 1 ) ; see also Lemma A.1 in the Appendix. However, since Λ ⁎ is in general not injective, the nonlinear system s = Λ ⁎ ( Λ ( x ) − 1 ) cannot be solved for x in closed form; only the x → s map is easily computable, not the converse. The same mapping − g has also been recently studied by Lasserre (2022) and others Castro et al. (2021). We shall elaborate more on this connection in Section 5. The following proposition gives two sufficient, although not necessary, conditions for x ∈ Σ ⁎ to certify a polynomial t . It also reveals that C ( s ) and P ( x ) are full-dimensional cones, that is, they have a non-empty interior: every sufficiently small perturbation of the gradient certificate of s certifies every sufficiently small perturbation of s . Proposition 1.6 Davis and Papp, 2022, Theorem 2.4 and Corollary 2.5 Suppose that x ∈ Σ ⁎ and s = − g ( x ) . 1. Then x is a dual certificate for every polynomial t satisfying ‖ t − s ‖ x ⁎ ≤ 1 . 2. If y is a vector that satisfies the inequality ‖ x − y ‖ x < 1 2 , then y ∈ Σ ⁎ , and x certifies t = − g ( y ) . Two very detailed examples illustrating the concept of dual certificates, the gradient certificate, and the construction of explicit WSOS representations from dual certificates can be found in our previous work (Davis and Papp, 2022, Examples 2 and 3). Recall that the bit size of an integer y ∈ Z is defined as 1 + ⌈ log 2 ⁡ ( | y | + 1 ) ⌉ , and that the bit size of a vector y ∈ Z n can be bounded from above by n times the bit size of its the largest (in size) component. As we are interested in the orders of magnitude of bit sizes of dual certificates (e.g., whether they are linear or polynomial or exponential functions of parameters such as the degree or the number of variables of the certified polynomials), it will be convenient but equally informative to substitute this quantity with the simpler log ⁡ ( ‖ y ‖ ∞ ) . Section snippets Rational certificates with boundable bit bize The goal of this section is to bound the norm of an integer dual certificate y ‾ ∈ Σ ⁎ of a polynomial t ∈ Σ ∘ . We consider different bounds, some of which depend only on the number of variables n , the degree d , and t , and others that are expressed in terms of other computable or interpretable parameters introduced later in this section. The strategy to derive these bounds is as follows. In Section 2.1, we show that dual certificates suitably close to the gradient certificate can be rounded to nearby Bit size bounds for rational certificates in particular bases In this section we refine the result of Theorem 2.9 in a few well-studied and computationally relevant special cases such as the cones of univariate polynomials nonnegative on the real line or on a bounded interval. These results complement existing ones on the bit sizes of conventional sum-of-squares certificates of nonnegative univariate polynomials, such as those summarized in the Introduction. We emphasize that our approach yields an efficiently computable bound for a variety of WSOS cones Computing certified WSOS lower bounds in rational arithmetic Having established the existence of rational dual certificates with a priori bounded bit sizes, we now turn to the question of computing such certificates. More precisely, given a polynomial t and a tolerance ε > 0 , we want to compute a rational lower bound c that lies between the optimal WSOS lower bound c ⁎ and c ⁎ − ε , along with a rational dual certificate (of a small bit size) proving t − c 1 ∈ Σ . The new algorithm (Algorithm 1 below) is an adaptation of Algorithm 1 from Davis and Papp (2022), which Discussion Bit size bounds on certificates from Algorithm 1 We opted to separate the discussion on the bit sizes of the certificates and Algorithm 1. In principle, one could study the former question “constructively” by analyzing the bit sizes of the certificates computed by the algorithm, but we think it is useful to underline that both the concept of dual certificates and the bit size bounds are independent of any particular algorithm. Theorem 2.4 and Lemma 2.8 are both derived assuming that the dual Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements The authors are grateful for the referees' thoughtful comments on the presentation, helping to make the paper more accessible to the symbolic computing and computational real algebraic geometry audience. DP would also like to thank Didier Henrion for pointing out the connection between gradient certificates and the Christoffel-Darboux polynomial, and Ali Mohammad Nezhad for pointing us to the results on the bit sizes of subresultants used in the proof of Corollary 3.2. References (27) V. Magron et al. On exact Reznick, Hilbert-Artin and Putinar's representations J. Symb. Comput. (2021) V. Magron et al. Algorithms for weighted sum of squares decomposition of non-negative univariate polynomials J. Symb. Comput. (2019) D. Papp Duality of sum of nonnegative circuit polynomials and optimal SONC bounds J. Symb. Comput. (2023) S. Basu et al. Algorithms in Real Algebraic Geometry (2006) S. Basu et al. A bound on the minimum of a real positive polynomial over the standard simplex B. Beckermann The condition number of real Vandermonde, Krylov and positive definite Hankel matrices Numer. Math. (2000) F. Boudaoud et al. Certificates of positivity in the Bernstein basis Discrete Comput. Geom. (2008) S.P. Boyd et al. Convex Optimization (2004) Y.D. Castro et al. Dual optimal design and the Christoffel–Darboux polynomial Optim. Lett. (2021) M.M. Davis et al. Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets SIAM J. Optim. (2022) G.H. Hardy et al. Inequalities (1934) L. Katthän et al. A unified framework of SAGE and SONC polynomials and its duality theory Math. Comput. 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