Powers (Pacific J. Math. 251 (2011), 385–391) proved that when a semialgebraic set and a strictly positive polynomial on it are given with rational coefficients, Schmüdgen-type representations can always be taken with sums of squares of polynomials over Q\mathbb{Q}, and the same holds for Putinar-type representations as long as a ball polynomial N−∑Xi2N-\sum X_i^2 is included among the generators. She asked whether this proviso is dispensable: if S⊆Q[X1,…,Xn]S\subseteq\mathbb{Q}[X_1,\dots,X_n] is finite and the quadratic module MSM_S is archimedean in R[X1,…,Xn]\mathbb{R}[X_1,\dots,X_n], is MSM_S then also archimedean as a quadratic module in Q[X1,…,Xn]\mathbb{Q}[X_1,\dots,X_n] — concretely, can the certificate of compactness N−∑Xi2N-\sum X_i^2 always be rewritten with rational sums-of-squares multipliers? This is the surviving open core of the rationality-transfer questions in that note: the companion question of Sturmfels, whether a rational polynomial that is a sum of squares over R\mathbb{R} is also a sum of squares over Q\mathbb{Q}, was answered negatively by Scheiderer (2016), and even strictly positive counterexamples were later constructed by Laplagne (2024). The archimedean descent question remains open in general; the only partial result known is the affirmative answer of Keshari–Ojha–Patra (2025) for the case where the support algebra Q[X]/(MS∩−MS)\mathbb{Q}[X]/(M_S\cap -M_S) is zero-dimensional.

Background

A polynomial that is nonnegative on all of Rn\mathbb{R}^n can sometimes be certified as such by writing it as a finite sum of squares (SOS) of polynomials. More generally, for a finite set S={g1,…,gs}S=\{g_1,\dots,g_s\} of polynomials one considers the basic closed semialgebraic set KS={x∈Rn:g1(x)≥0,…,gs(x)≥0}K_S=\{x\in\mathbb{R}^n:g_1(x)\ge 0,\dots,g_s(x)\ge 0\}, and an explicit representation of a polynomial ff as an SOS-weighted combination of the gig_i certifies that ff is nonnegative on KSK_S. Two representation theorems are central. The preordering TST_S consists of all finite sums ∑e∈{0,1}sσeg1e1⋯gses\sum_{e\in\{0,1\}^s}\sigma_e g_1^{e_1}\cdots g_s^{e_s} with each σe\sigma_e an SOS; Schmüdgen's theorem (1991) states that if KSK_S is compact, then every polynomial strictly positive on KSK_S lies in TST_S. The quadratic module MSM_S consists of all f=σ0+σ1g1+⋯+σsgsf=\sigma_0+\sigma_1g_1+\cdots+\sigma_sg_s with SOS multipliers σi\sigma_i (each constraint polynomial used at most once); Putinar's theorem (1993) draws the same conclusion under the stronger hypothesis that MSM_S is archimedean, meaning that every polynomial is bounded by some integer in MSM_S — equivalently, that some ball polynomial N−∑iXi2N-\sum_iX_i^2 belongs in MSM_S, so that the ball polynomial serves as a certificate of compactness for KSK_S. These theorems underlie the standard semidefinite-programming hierarchy for polynomial optimization. Since the underlying solvers work in floating point, the certificates they output are numerical, and the exact-certification question arises: if SS and ff have rational coefficients, do representations with rational coefficients always exist? Powers (2011) proved that the answer is yes for Schmüdgen representations on compact KSK_S, and yes for Putinar representations whenever a ball polynomial is available among the generators; her arguments descend, respectively, a purely algebraic proof of Schmüdgen's theorem (via the abstract Positivstellensatz for preordered rings) and an algorithmic proof of Putinar's theorem (via Pólya's classical theorem that a homogeneous polynomial strictly positive on the positive orthant acquires nonnegative coefficients after multiplication by a power of the sum of its variables) to rational coefficients. Her note leaves open whether the archimedean hypothesis itself descends from R\mathbb{R} to Q\mathbb{Q} — the question of this record. The subtlety is real: for the companion question raised by Sturmfels, whether an f∈Q[X1,…,Xn]f\in\mathbb{Q}[X_1,\dots,X_n] that is an SOS in R[X1,…,Xn]\mathbb{R}[X_1,\dots,X_n] must be an SOS in Q[X1,…,Xn]\mathbb{Q}[X_1,\dots,X_n], the answer is negative. Hillar (2009) had proved the positive transfer under the extra hypothesis that ff is a sum of squares over a totally real number field (a number field all of whose complex embeddings are real), with an explicit bound on the number of rational squares needed. But Scheiderer (2016) constructed explicit rational polynomials that are sums of squares over R\mathbb{R} yet not over Q\mathbb{Q} — norm forms, i.e. products of all Galois conjugates, of linear forms over totally imaginary number fields (number fields with no real embedding) whose Galois group acts sufficiently transitively on the complex embeddings — and Laplagne (2024) produced such an example that is even strictly positive everywhere. In the plain SOS setting the obstruction is the possible absence of rational points on the boundary of the Gram spectrahedron (the convex set of symmetric matrices AA with f=v⊤Avf=v^{\top}Av, where vv is the vector of monomials, in which a rational SOS decomposition corresponds to a rational positive semidefinite Gram matrix). For the archimedean question, the analogous issue is whether the semialgebraic set of tuples of rational SOS multipliers representing a ball polynomial can be empty while the corresponding set over R\mathbb{R} is nonempty — precisely the failure mode that the known SOS counterexamples realize for single representations, but which has not been shown possible for the archimedean property itself.

Problem Statement

Let n≥1n\ge 1 and let S={g1,…,gs}⊆Q[X1,…,Xn]S=\{g_1,\dots,g_s\}\subseteq\mathbb{Q}[X_1,\dots,X_n] be a finite set of polynomials with rational coefficients. For a field F⊆RF\subseteq\mathbb{R}, write ∑F[X]2\sum F[X]^2 for the set of sums of squares of polynomials with coefficients in FF, and let MS(F)={σ0+σ1g1+⋯+σsgs: σi∈∑F[X]2}M_S(F)=\{\sigma_0+\sigma_1g_1+\cdots+\sigma_sg_s:\ \sigma_i\in\textstyle\sum F[X]^2\} denote the quadratic module generated by SS in F[X1,…,Xn]F[X_1,\dots,X_n]. Suppose that MS(R)M_S(\mathbb{R}) is archimedean, that is, every element of R[X1,…,Xn]\mathbb{R}[X_1,\dots,X_n] is bounded in MS(R)M_S(\mathbb{R}); equivalently, there exist N∈NN\in\mathbb{N} and real sums of squares σi∈∑R[X]2\sigma_i\in\sum\mathbb{R}[X]^2 with N−∑i=1nXi2=σ0+σ1g1+⋯+σsgs.N-\sum_{i=1}^nX_i^2=\sigma_0+\sigma_1g_1+\cdots+\sigma_sg_s. Question (Powers, 2011): must MS(Q)M_S(\mathbb{Q}) then also be archimedean? Concretely, must there exist N′∈NN'\in\mathbb{N} and rational sums of squares τi∈∑Q[X]2\tau_i\in\sum\mathbb{Q}[X]^2 with N′−∑i=1nXi2=τ0+τ1g1+⋯+τsgs,N'-\sum_{i=1}^nX_i^2=\tau_0+\tau_1g_1+\cdots+\tau_sg_s, or, equivalently, some N′′∈NN''\in\mathbb{N} such that each N′′±XjN''\pm X_j admits such a representation with rational sums-of-squares multipliers? The question is posed at this full generality — all nn and all finite S⊆Q[X1,…,Xn]S\subseteq\mathbb{Q}[X_1,\dots,X_n], with no restriction on the geometry of KSK_S; in particular, the typical case where KSK_S has nonempty interior in Rn\mathbb{R}^n is included and is not covered by the known partial results. A complete answer is either a proof that archimedeanness of a finitely generated quadratic module always descends from R[X1,…,Xn]\mathbb{R}[X_1,\dots,X_n] to Q[X1,…,Xn]\mathbb{Q}[X_1,\dots,X_n], or an explicit finite set SS of rational polynomials together with a proof that MS(R)M_S(\mathbb{R}) is archimedean while MS(Q)M_S(\mathbb{Q}) is not.

References

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