Descent of nonnegativity certificates to rational coefficients: finite generation of $Pos(K)$ as a preordering of $\mathbb{Q}[x]$ for specific compact intervals with algebraic boundary, and archimedeanity of rational quadratic modules

For a basic closed semialgebraic set $K\subset\mathbb{R}$ defined over $\mathbb{Q}$, every polynomial nonnegative on $K$ admits a sum-of-squares certificate with real coefficients, but it is open when such certificates exist with rational coefficients and finitely many generators. The problem asks, for specific compact intervals $K$ whose boundary points are algebraic but not rational — $[0,2^{1/3}]$, $[\sqrt{2},\sqrt{3}]$, and $[-\sqrt[4]{2},\sqrt[4]{2}]$ — whether $Pos(K)$ (the set of polynomials nonnegative on $K$) is a finitely generated preordering of $\mathbb{Q}[x]$; the source's membership probes — whether $x-1\in T\{x,x^{2}-2,3-x^{2}\}$ and whether $x-\alpha\in T\{2-x^{4}\}$ for rational $\alpha\in(\sqrt[4]{2},\sqrt{2})$ — are each settled by an immediate argument (see the Problem Statement). It further asks for uniform degree bounds on certificates when saturation of the natural quadratic module fails, and whether archimedeanity of a quadratic module over $\mathbb{R}$ descends to $\mathbb{Q}$ without the zero-dimensional support hypothesis. These are the open descent questions left by Keshari, Ojha and Patra (arXiv:2508.07060, 2025), partially answering a question of Powers (2011).

Background

Let $F$ be a subfield of $\mathbb{R}$ and $S=\{g_{1},\dots,g_{s}\}\subset F[x]$. The preordering $T_{S}$ is the smallest subset of $F[x]$ containing $S$ and all sums of squares of polynomials that is closed under addition and multiplication; its elements are finite sums $\sum_{e}\sigma_{e}\prod_{i}g_{i}^{e_{i}}$ with $\sigma_{e}$ sums of squares. The quadratic module $M_{S}=\{\sigma_{0}+\sigma_{1}g_{1}+\dots+\sigma_{s}g_{s}:\sigma_{i}\text{ sums of squares}\}$ is the analogous additive-only structure. Both define certificates of nonnegativity on the set $K_{S}=\{x\in\mathbb{R}:g_{i}(x)\ge0\}$, since every element of $T_{S}$ or $M_{S}$ is visibly nonnegative on $K_{S}$. Write $Pos(K)$ for the set of polynomials nonnegative at every point of $K$; $T_{S}$ (resp. $M_{S}$) is called saturated when it equals $Pos(K_{S})$, so that every nonnegative polynomial is certified. A quadratic module is archimedean when it contains $N-\sum_{i}x_{i}^{2}$ for some positive integer $N$; by Putinar's theorem archimedeanity guarantees saturation of quadratic modules for strictly positive polynomials over $\mathbb{R}$, and Schmüdgen's theorem gives saturation of preorderings on compact sets over $\mathbb{R}$. For a finite union of closed intervals $K\neq\mathbb{R}$ with boundary $\partial K\subset F$, Kuhlmann and Marshall introduced the natural generators $\mathcal{N}\subset F[x]$: the linear polynomial $x-a$ (resp. $a-x$) for each boundary point $a$ with $(-\infty,a)\cap K=\emptyset$ (resp. $(a,\infty)\cap K=\emptyset$), and the quadratic $(x-a)(x-b)$ for each pair of boundary points $a<b$ with $(a,b)\cap K=\emptyset$; for $K=\mathbb{R}$ one takes $\mathcal{N}=\{1\}$. They proved over $\mathbb{R}$ that $T_{\mathcal{N}}$ is saturated; Keshari, Ojha and Patra (2025) proved that this saturation descends to every subfield $F$, i.e. $T_{\mathcal{N}}=Pos(K)$ in $F[x]$ whenever $\partial K\subset F$. Scheiderer (2016) analyzed when sums of squares with real coefficients can be rewritten with rational (or number-field) coefficients, giving both positive descent results and counterexamples. The descent problem, posed in the archimedean case by Powers (2011), asks which of these certificate structures survive when coefficients are restricted from $\mathbb{R}$ to $\mathbb{Q}$: saturation can fail because $\mathbb{Q}$ misses the algebraic boundary points, and finite generation of $Pos(K)$ as a preordering of $\mathbb{Q}[x]$ becomes a delicate question of field arithmetic at each boundary point. This matters for exact polynomial optimization and computer algebra, where certificates over $\mathbb{Q}$ can be checked by exact rational arithmetic.

Problem Statement

All rings are $\mathbb{Q}[x]$ (in item (C), $\mathbb{Q}[x_{1},\dots,x_{n}]$) and all sets $K$ are basic closed semialgebraic subsets of $\mathbb{R}$ defined over $\mathbb{Q}$. Recall from the background that $K\{g_{1},\dots,g_{s}\}$ denotes the basic closed set $K_{S}$ cut out by $S=\{g_{1},\dots,g_{s}\}$, that $T\{g_{1},\dots,g_{s}\}$ denotes the preordering generated by them, and that $\mathcal{N}$ is the natural choice of generators. Answer the following open questions of Keshari, Ojha and Patra (arXiv:2508.07060). (A) Question 8.8: is $Pos(K)$ a finitely generated preordering of $\mathbb{Q}[x]$ in the following compact cases with $\partial K\not\subset\mathbb{Q}$ — (1) $K=[0,2^{1/3}]=K\{x,2-x^{3}\}$; (2) $K=[\sqrt{2},\sqrt{3}]=K\{x,x^{2}-2,3-x^{2}\}$, in particular does $x-1\in T\{x,x^{2}-2,3-x^{2}\}$; (3) $K=[-\sqrt[4]{2},\sqrt[4]{2}]=K\{2-x^{4}\}$, in particular does $x-\alpha\in T\{2-x^{4}\}$ for some or all $\alpha\in\mathbb{Q}\cap(\sqrt[4]{2},\sqrt{2})$? Here $\sqrt[4]{2}$ and $\sqrt{2}$ denote the positive real fourth and square roots of 2. Note that $x-1$ is strictly positive on the compact set $K=[\sqrt{2},\sqrt{3}]$, so the membership question in (2) is answered in the affirmative by Schmüdgen's theorem together with its descent to $\mathbb{Q}$ (Powers 2011); and for every such $\alpha$ one has $x-\alpha<0$ at $x=\sqrt[4]{2}\in K$, while every element of $T\{2-x^{4}\}$ is nonnegative on $K$, so the membership question in (3) is answered in the negative by this sign observation. The open content of (1), (2) and (3) is therefore the finite generation of $Pos(K)$ as a preordering of $\mathbb{Q}[x]$. (B) Question 6.11(2) in the unsolved regime: when $K$ is non-compact and either (i) the natural generator set $\mathcal{N}=\{g_{1},\dots,g_{s}\}$ has $s=2$ generators and $K$ has no isolated point, or (ii) $s\ge3$ generators — the regime where $M_{\mathcal{N}}\ne T_{\mathcal{N}}$ — does every $f\in M_{\mathcal{N}}$ admit a representation $f=\sigma_{0}+\sigma_{1}g_{1}+\dots+\sigma_{s}g_{s}$ with $\sigma_{i}\in\sum\mathbb{Q}_{\ge0}\mathbb{Q}[x]^{2}$ such that $\deg(\sigma_{i}g_{i})$ is bounded by a function of $\deg(f)$? (C) Powers' question: if $S\subset\mathbb{Q}[x_{1},\dots,x_{n}]$ and $M_{S}\otimes\mathbb{R}$ is archimedean, must $M_{S}$ be archimedean? Theorem 9.3 of the source settles this only under the hypothesis $\dim(\mathbb{Q}[x_{1},\dots,x_{n}]/(M_{S}\cap-M_{S}))=0$; the positive-dimensional case is open.

References

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