Putinar's Positivstellensatz (1993) represents every polynomial that is strictly positive on a compact basic semialgebraic set as a sum-of-squares combination of the defining constraint polynomials, provided the associated quadratic module is Archimedean - but with real coefficients. When the constraints and the polynomial have coefficients in a subfield K⊂RK\subset\mathbb{R} (the central case being K=QK=\mathbb{Q}), it is an open problem whether the certificate can always be chosen with coefficients in KK itself. The statement is known only under additional hypotheses: an explicit ball modulus over KK (Powers, 2011), a unisolvent property implied by nonempty interior of the set (Davis-Papp, 2022), zero-dimensional (finite) constraint ideals (Baldi-Krick-Mourrain, the source of this problem), and gradient-ideal cases (Magron-Safey El Din-Vu, 2023). The standard proof route passes through Powers' question of whether the Archimedean property descends from the real quadratic module to the subfield module; Keshari-Ojha-Patra (2025) answered that question only when the support of the module is zero-dimensional, i.e. essentially the finite-set case. The problem asks for a proof or refutation of the field-preserving Putinar statement - equivalently, of the Archimedean descent - for positive-dimensional compact semialgebraic sets, where all known sufficient conditions fail.

Background

A basic closed semialgebraic set is a set of the form S={ξ∈Rn:g1(ξ)≥0,…,gr(ξ)≥0}S=\{\xi\in\mathbb{R}^n: g_1(\xi)\ge 0,\dots,g_r(\xi)\ge 0\} defined by finitely many polynomial inequalities. A polynomial is a sum of squares (SoS) if it is a finite sum of squares of polynomials; over R[x1,…,xn]\mathbb{R}[x_1,\dots,x_n], being an SoS is equivalent to positive semidefiniteness of an associated Gram matrix, which connects the theory to semidefinite programming (SDP), the optimization of linear objectives over positive semidefinite matrices. Given constraints g1,…,grg_1,\dots,g_r, the quadratic module they generate is the set Q={σ0+∑i=1rσigi}Q=\{\sigma_0+\sum_{i=1}^r \sigma_i g_i\} where the σi\sigma_i are SoS; the larger preordering additionally allows products of the gig_i as multipliers. When working over a proper subfield K⊂RK\subset\mathbb{R} one uses weighted sums of squares ΣK2\Sigma_K^2, in which each square is multiplied by a nonnegative weight ω∈K≥0\omega\in K_{\ge 0}; this refinement is necessary because not every nonnegative element of a subfield is a sum of squares of elements of that field (for instance 2∈Q(2)\sqrt{2}\in\mathbb{Q}(\sqrt{2}) is not), while over Q\mathbb{Q} and R\mathbb{R} weighted and ordinary SoS coincide. A quadratic module is Archimedean if it contains a ball modulus N−∑ixi2N-\sum_i x_i^2 for some NN (equivalently, a polynomial whose nonnegativity set is compact); in particular its associated set SS is then compact.

Putinar's Positivstellensatz states that if QRQ_{\mathbb{R}} is Archimedean and ff is strictly positive on SS, then f∈QRf\in Q_{\mathbb{R}}; Schmüdgen's theorem states that compactness of SS alone suffices for membership in the preordering. Both conclusions a priori involve real coefficients, even when the input data are rational.

The coefficient field matters for computation: SDP solvers work in floating point, and converting a numerical certificate into an exact, machine-checkable one requires the certificate's coefficients to lie in Q\mathbb{Q} (or a similarly explicit field); exact rational SoS certificates are also what is needed for positivity witnesses in theorem provers and for bit-complexity analysis of the moment/SoS hierarchy used in polynomial optimization.

The descent problem asks which of these statements remain true when R\mathbb{R} is replaced by a subfield K⊂RK\subset\mathbb{R}. Schmüdgen's statement descends: it holds over any Archimedean subfield (via Schweighofer's algorithmic proof). For Putinar's statement only partial results are known. Powers proved descent under the explicit assumption that a ball modulus R−∑ixi2R-\sum_i x_i^2 with R∈K≥0R\in K_{\ge 0} already lies in the module over KK, and posed the question whether Archimedeanity over Q\mathbb{Q} implies that condition. Davis and Papp proved rational SoS decompositions for polynomial optimization over compact sets under a unisolvent property of the constraint set, which is implied by nonemptiness of the Euclidean interior of SS. Magron, Safey El Din and Vu treated the case where the representation is taken modulo the gradient ideal of ff (the ideal generated by the partial derivatives of ff) under a zero-dimensionality assumption. Baldi, Krick and Mourrain - the source of this problem - settled the zero-dimensional case completely: for a finite basic semialgebraic set defined by equality constraints generating a zero-dimensional ideal II (i.e. the complex solution set of the equalities is finite, so K[x]/IK[x]/I is a finite-dimensional vector space), possibly with sign constraints, every strictly positive f∈K[x]f\in K[x] admits a field-preserving weighted-SoS representation, with degree bounds linear in the Castelnuovo-Mumford regularity (a standard homological complexity measure of the quotient algebra) when the equalities form a graded basis, and, over Q\mathbb{Q}, bit-size bounds quadratic or cubic in the Bézout bound dnd^n on the number of complex solutions; as an application they retrieve and improve finite-convergence results for the moment/SoS hierarchy in discrete optimization. They state that this is, to their knowledge, the first general Putinar-type representation over the rationals, and that the extension beyond finite sets is open.

Two facts delimit the question. First, descent genuinely fails without compactness or strict positivity: Scheiderer exhibited polynomials nonnegative on all of Rn\mathbb{R}^n with rational coefficients that are real sums of squares but not rational ones, and in the zero-dimensional setting even nonnegative (as opposed to strictly positive) polynomials can lack field-preserving representations. Second, the known positive cases all exploit a strict-feasibility margin: a real Putinar certificate corresponds to positive semidefinite Gram matrices, and rounding its real entries into the subfield preserves the representation identity only when the matrices are strictly feasible (Peyrl-Parrilo rounding; Davis-Papp dual certificates; the zero-dimensional quotient algebra of Baldi-Krick-Mourrain); on the boundary of the cone of positive semidefinite matrices such perturbation can fail. The alternative route - proving that the Archimedean property itself descends from QRQ_{\mathbb{R}} to QKQ_K - is Powers' question, recently studied systematically by Keshari, Ojha and Patra, who confirmed it only when the support ideal QK∩(−QK)Q_K\cap(-Q_K) has a zero-dimensional quotient (essentially the finite-set case), using semi-orderings (a relaxation of orderings - subsets of the ring closed under addition and multiplication, containing all squares and, for each aa, at least one of a,−aa,-a; both may lie in a semi-ordering, unlike the exactly-one-of-a,−aa,-a condition that defines an ordering - from the general theory of quadratic modules over ordered fields). What remains untouched is precisely the positive-dimensional case, in particular compact sets with empty interior such as curve or surface segments, where neither the interior-based nor the zero-dimensional arguments apply.

Problem Statement

Let K⊂RK\subset\mathbb{R} be a subfield (the central case is K=QK=\mathbb{Q}), let g1,…,gr∈K[x1,…,xn]g_1,\dots,g_r\in K[x_1,\dots,x_n], let

S={ξ∈Rn:gi(ξ)≥0 for 1≤i≤r},S=\{\xi\in\mathbb{R}^n : g_i(\xi)\ge 0 \text{ for } 1\le i\le r\},

and let

QK=ΣK2+∑i=1rΣK2⋅gi ⊂ K[x1,…,xn]Q_K=\Sigma_K^2+\sum_{i=1}^r \Sigma_K^2\cdot g_i\ \subset\ K[x_1,\dots,x_n]

be the quadratic module generated by the gig_i over KK, where ΣK2\Sigma_K^2 denotes the cone of weighted sums of squares with weights in K≥0K_{\ge 0}. Let QR⊂R[x1,…,xn]Q_{\mathbb{R}}\subset\mathbb{R}[x_1,\dots,x_n] be the corresponding quadratic module over R\mathbb{R}. Assume QRQ_{\mathbb{R}} is Archimedean, that is, N−∑i=1nxi2∈QRN-\sum_{i=1}^n x_i^2\in Q_{\mathbb{R}} for some integer NN; in particular SS is compact.

Decide the following.

(1) Field-preserving Putinar (Putinar descent): does every f∈K[x1,…,xn]f\in K[x_1,\dots,x_n] with f>0f>0 on SS admit a representation

f=σ0+∑i=1rσi gi,σi∈ΣK2?f=\sigma_0+\sum_{i=1}^r \sigma_i\, g_i,\qquad \sigma_i\in\Sigma_K^2?

(2) Archimedean descent (Powers' question, a positive answer to which implies (1) via Powers' theorem): does N−∑ixi2∈QRN-\sum_i x_i^2\in Q_{\mathbb{R}} for some N∈NN\in\mathbb{N} imply that N′−∑ixi2∈QKN'-\sum_i x_i^2\in Q_K for some N′∈NN'\in\mathbb{N}?

A positive answer must hold for all subfields K⊂RK\subset\mathbb{R}, or at least for K=QK=\mathbb{Q}, with no hypotheses beyond those stated - in particular without assuming that SS has nonempty interior, that the constraints generate a zero-dimensional ideal, that an explicit ball modulus is available over KK, or that the representation may be taken modulo any auxiliary ideal. A negative answer may be given by a single explicit instance: a subfield KK (in particular K=QK=\mathbb{Q} is admissible), explicit constraints g1,…,grg_1,\dots,g_r with QRQ_{\mathbb{R}} Archimedean, and either an explicit f∈K[x1,…,xn]f\in K[x_1,\dots,x_n], strictly positive on SS, that provably admits no representation as in (1), or a proof that QKQ_K is not Archimedean.

References

  1. L. Baldi, T. Krick, B. Mourrain, An effective Positivstellensatz over the rational numbers for finite semialgebraic sets, Mathematics of Computation (2025), DOI 10.1090/mcom/4153; arXiv:2410.04845, https://arxiv.org/abs/2410.04845
  2. M. K. Keshari, D. Ojha, N. S. Patra, Descent problem for certificate of non-negativity on semi-algebraic sets, arXiv:2508.07060 (2025), https://arxiv.org/abs/2508.07060
  3. V. Powers, Rational certificates of positivity on compact semialgebraic sets, Pacific Journal of Mathematics 251(2):385-391 (2011), DOI 10.2140/pjm.2011.251.385, https://doi.org/10.2140/pjm.2011.251.385
  4. M. Putinar, Positive polynomials on compact semi-algebraic sets, Indiana University Mathematics Journal 42(3):969-984 (1993), DOI 10.1512/iumj.1993.42.42045, https://doi.org/10.1512/iumj.1993.42.42045
  5. K. Schmüdgen, The K-moment problem for compact semi-algebraic sets, Mathematische Annalen 289(1):203-206 (1991), DOI 10.1007/BF01446568, https://doi.org/10.1007/BF01446568
  6. M. Schweighofer, An algorithmic approach to Schmüdgen's Positivstellensatz, Journal of Pure and Applied Algebra 166(3):307-319 (2002), DOI 10.1016/S0022-4049(01)00041-X, https://doi.org/10.1016/S0022-4049(01)00041-X
  7. C. Scheiderer, Sums of squares of polynomials with rational coefficients, Journal of the European Mathematical Society 18(7):1495-1513 (2016), DOI 10.4171/jems/620, https://doi.org/10.4171/jems/620
  8. J. Capco, C. Scheiderer, Two remarks on sums of squares with rational coefficients, Banach Center Publications 121:25-36 (2020), DOI 10.4064/bc121-2, https://doi.org/10.4064/bc121-2
  9. M. M. Davis, D. Papp, Dual certificates and efficient rational sum-of-squares decompositions for polynomial optimization over compact sets, SIAM Journal on Optimization 32(4):2461-2492 (2022), DOI 10.1137/21m1422574, https://doi.org/10.1137/21m1422574
  10. M. M. Davis, D. Papp, Rational dual certificates for weighted sums-of-squares polynomials with boundable bit size, Journal of Symbolic Computation 121:102254 (2024), DOI 10.1016/j.jsc.2023.102254, https://doi.org/10.1016/j.jsc.2023.102254
  11. H. Peyrl, P. A. Parrilo, Computing sum of squares decompositions with rational coefficients, Theoretical Computer Science 409(2):269-281 (2008), DOI 10.1016/j.tcs.2008.09.025, https://doi.org/10.1016/j.tcs.2008.09.025
  12. V. Magron, M. Safey El Din, T.-H. Vu, Sum of squares decompositions of polynomials over their gradient ideals with rational coefficients, SIAM Journal on Optimization 33(1):63-88 (2023), DOI 10.1137/21m1436245, https://doi.org/10.1137/21m1436245
  13. S. Kuhlmann, M. Marshall, Positivity, sums of squares and the multi-dimensional moment problem, Transactions of the American Mathematical Society 354(11):4285-4301 (2002), DOI 10.1090/s0002-9947-02-03075-1, https://doi.org/10.1090/s0002-9947-02-03075-1
  14. M. R. Bender, K. Kozhasov, E. P. Tsigaridas, C. Zhu, Certificates for nonnegativity of multivariate integer polynomials under perturbations, arXiv:2512.09808 (2025), https://arxiv.org/abs/2512.09808
  15. J. A. Castellanos-Joo, D. Kapur, Computing certificates in Archimedean univariate saturated quadratic modules, arXiv:2605.18980 (2026), https://arxiv.org/abs/2605.18980