arXiv subjects

Lorenzo Baldi

Publications and source records attributed to Lorenzo Baldi.

12 recordsLinked to original sources

arXiv2026

A note on bounded ratios

We prove that the set of bounded ratios $\BR(X)$ on a semialgebraic set $X\subset\R^n_{>0}$ is the convex cone of linear forms that are nonnegative on the tropicalization $\trop(X)$. In particular, it is a rational polyhedral convex cone. For $X$ the set of Lorentzian polynomials with fixed M-convex support, it is the dual to the set of M-convex functions. We record an explicit counterexample to a conjecture of Huang--Huh--Soskin--Wang on the bounded ratios on Lorentzian polynomials. The bounded ratio in the counterexample corresponds to the non-hypermetric clique-web facet $\mathrm{CW}^1_7(1,1,1,1,1,-1,-1)$ of the cut cone on seven vertices.

math.CO↗

arXiv2026

A note on o-minimal tropicalizations

In this note, we provide several equivalent descriptions of the tropicalization of definable sets in a polynomially bounded o-minimal expansion of a real closed field. We show that the tropicalization can be described in multiple ways: as the image under the valuation map, as the image under the tropicalization map of the archimedean points in the o-minimal spectrum, in terms of initial degenerations of the definable set and initial nonnegativity cones, or via the real tropicalization of polynomials belonging to the corresponding nonnegativity cone. Moreover, we prove analogous results for the fine tropicalization of definable sets, where the valuation map is replaced by the RV sort map, which records not only the valuation of a point but also its angular component.

math.AG↗

arXiv2026

Stubborn Polynomials

The relationship between nonnegative polynomials and sums of squares is a classical topic in real algebraic geometry. We study \emph{stubborn polynomials} $f$ on a real variety $X$, which are polynomials nonnegative on $X$, such that no odd power of $f$ is a sum of squares. Previously, stubborn polynomials were studied only in the globally nonnegative case, with results restricted to polynomials nonnegative on $\mathbb{P}^2$. We fully characterize stubborn polynomials on smooth curves, showing that a polynomial on a smooth totally real curve is stubborn if and only if all of its zeros are real. This implies that there exist smooth curves with no stubborn polynomials in low degree, while stubborn polynomials must exist in sufficiently high degrees on curves with positive genus. We explore the much more delicate situation with singular and reducible curves. While being real-rooted always implies being stubborn, there also exist singular curves with no stubborn polynomials at all. We then analyze the case of ternary sextics, i.e.,~polynomials of degree $6$ on $\mathbb{P}^2$. We prove the Conjecture of Blekherman, Kozhasov, and Reznick that a nonnegative ternary sextic is stubborn if and only if its real delta-invariant is at least 9. To analyze this case, we develop results for lifting stubborn polynomials from curves to higher dimensional varieties and use the theory of weak Del Pezzo surfaces. We complement these results with structural properties of stubborn polynomials and present many explicit examples.

math.AG↗

arXiv2025

Toric extensions of P\'olya's theorem

The classical version of P\'olya's theorem provides a simple method for certifying that a homogeneous polynomial of degree d is strictly copositive, that is, it takes only positive values on the nonnegative real orthant. However, this method might fail to detect copositivity of polynomials that are missing certain degree d monomials. In this paper, we present extensions and converses to P\'olya's theorem for sparse polynomials, using techniques from positive toric geometry. Furthermore, we explore how this method can be used to study the convergence of Feynman integrals in particle physics.

math.AG↗

arXiv2025

Totally real divisors on curves

Since the works of Krasnov and Scheiderer, there has been an interest in studying effective totally real divisors on a curve X defined over a real closed field, i.e., effective divisors supported on the real locus. Scheiderer proved that, for smooth curves over the real numbers with nonempty real locus, each divisor of sufficiently high degree is linearly equivalent to an effective totally real one. The smallest degree N(X) with this property is called the totally real divisor threshold. When the field is non-Archimedean, we obtain a classification of topological types of smooth curves for which N(X) can be infinite. As a consequence, for curves over the real numbers we prove that N(X) cannot be bounded from above only in terms of the topological type, unless the real locus has many connected components. We complement this qualitative result with a quantitative lower bound for N(X), depending on metric properties of the Jacobian and the curve in the Bergman metric. Finally, we relate these metric properties to period matrices of X, expressed in a way compatible with the real structure.

math.AG↗

arXiv2024

An Effective Positivstellensatz over the Rational Numbers for Finite Semialgebraic Sets

We study the problem of representing multivariate polynomials with rational coefficients, which are nonnegative and strictly positive on finite semialgebraic sets, using rational sums of squares. We focus on the case of finite semialgebraic sets S defined by equality constraints, generating a zero-dimensional ideal I, and by nonnegative sign constraints. First, we obtain existential results. We prove that a strictly positive polynomial f with coefficients in a subfield K of R has a representation in terms of weighted Sums-of-Squares with coefficients in this field, even if the ideal I is not radical. We generalize this result to the case where f is nonnegative on S and (f ) + (I : f ) = 1. We deduce that nonnegative polynomials with coefficients in K can be represented in terms of Sum-of-Squares of polynomials with coefficients in K, when the ideal is radical. Second, we obtain degree bounds for such Sums-of-Squares representations, which depend linearly on the regularity of the ideal and the degree of the defining equations, when they form a graded basis. Finally, we analyze the bit complexity of the Sums-of-Squares representations for polynomials with coefficients in Q, in the case the ideal is radical. The bitsize bounds are quadratic or cubic in the Bezout bound, and linear in the regularity, generalizing and improving previous results obtained for special zero dimensional ideals. As an application in the context of polynomial optimization, we retrieve and improve results on the finite convergence and exactness of the moment/Sums-of-Squares hierarchy.

math.AG↗

arXiv2024

Nonnegative Polynomials and Moment Problems on Algebraic Curves

The cone of nonnegative polynomials is of fundamental importance in real algebraic geometry, but its facial structure is understood in very few cases. We initiate a systematic study of the facial structure of the cone of nonnegative polynomials $\pos$ on a smooth real projective curve $X$. We show that there is a duality between its faces and totally real effective divisors on $X$. This allows us to fully describe the face lattice in case $X$ has genus one. We compute the Carath\'{e}odory number of the dual moment cone $\pos^\vee$ for an elliptic normal curve $X$, which measures the complexity of quadrature rules of measures supported on $X$. Interestingly, the topology of the real locus of $X$ influences the Carath\'{e}odory number of $\pos^\vee$. We apply our results to truncated moment problems on affine cubic curves, where we deduce sharp bounds on the flat extension degree.

math.AG↗

arXiv2023

Degree bounds for Putinar's Positivstellensatz on the hypercube

The Positivstellens\"atze of Putinar and Schm\"udgen show that any polynomial $f$ positive on a compact semialgebraic set can be represented using sums of squares. Recently, there has been large interest in proving effective versions of these results, namely to show bounds on the required degree of the sums of squares in such representations. These effective Positivstellens\"atze have direct implications for the convergence rate of the celebrated moment-SOS hierarchy in polynomial optimization. In this paper, we restrict to the fundamental case of the hypercube $\mathrm{B}^{n} = [-1, 1]^n$. We show an upper degree bound for Putinar-type representations on $\mathrm{B}^{n}$ of the order $O(f_{\max}/f_{\min})$, where $f_{\max}$, $f_{\min}$ are the maximum and minimum of $f$ on $\mathrm{B}^{n}$, respectively. Previously, specialized results of this kind were available only for Schm\"udgen-type representations and not for Putinar-type ones. Complementing this upper degree bound, we show a lower degree bound in $\Omega(\sqrt[8]{f_{\max}/f_{\min}})$. This is the first lower bound for Putinar-type representations on a semialgebraic set with nonempty interior described by a standard set of inequalities.

math.AG↗

arXiv2022

On {\L}ojasiewicz Inequalities and the Effective Putinar's Positivstellensatz

The representation of positive polynomials on a semi-algebraic set in terms of sums of squares is a central question in real algebraic geometry, which the Positivstellensatz answers. In this paper, we study the effective Putinar's Positivestellensatz on a compact basic semi-algebraic set $S$ and provide a new proof and new improved bounds on the degree of the representation of positive polynomials. These new bounds involve a parameter $\epsilon$ measuring the non-vanishing of the positive function, the constant $\mathfrak{c}$ and exponent $L$ of a {\L}ojasiewicz inequality for the semi-algebraic distance function associated to the inequalities $\mathbf{g} = (g_1, \dots , g_r)$ defining $S$. They are polynomial in $\mathfrak{c}$ and $\epsilon^{-1}$ with an exponent depending only on $L$. We analyse in details the {\L}ojasiewicz inequality when the defining inequalities $\mathbf g$ satisfy the Constraint Qualification Condition. We show that, in this case, the {\L}ojasiewicz exponent $L$ is $1$ and we relate the {\L}ojasiewicz constant $\mathfrak{c}$ with the distance of $\mathbf g$ to the set of singular systems.

math.AC↗

arXiv2021

On the Effective Putinar's Positivstellensatz and Moment Approximation

We analyse the representation of positive polynomials in terms of Sums of Squares. We provide a quantitative version of Putinar's Positivstellensatz over a compact basic semialgebraic set S, with a new polynomial bound on the degree of the positivity certificates. This bound involves a Lojasiewicz exponent associated to the description of S. We show that if the gradients of the active constraints are linearly independent on S (Constraint Qualification condition),this Lojasiewicz exponent is equal to 1. We deduce the first general polynomial bound on the convergence rate of the optima in Lasserre's Sum-of-Squares hierarchy to the global optimum of a polynomial function on S, and the first general bound on the Hausdorff distance between the cone of truncated (probability) measures supported on S and the cone of truncated pseudo-moment sequences, which are positive on the quadratic module of S.

math.AC↗

arXiv2021

Computing real radicals by moment optimization

We present a new algorithm for computing the real radical of an ideal and, more generally, the-radical of, which is based on convex moment optimization. A truncated positive generic linear functional vanishing on the generators of is computed solving a Moment Optimization Problem (MOP). We show that, for a large enough degree of truncation, the annihilator of generates the real radical of. We give an effective, general stopping criterion on the degree to detect when the prime ideals lying over the annihilator are real and compute the real radical as the intersection of real prime ideals lying over. The method involves several ingredients, that exploit the properties of generic positive moment sequences. A new efficient algorithm is proposed to compute a graded basis of the annihilator of a truncated positive linear functional. We propose a new algorithm to check that an irreducible decomposition of an algebraic variety is real, using a generic real projection to reduce to the hypersurface case. There we apply the Sign Changing Criterion, effectively performed with an exact MOP. Finally we illustrate our approach in some examples.

math.AC↗

arXiv2020

Exact Moment Representation in Polynomial Optimization

We investigate the problem of representing moment sequences by measures in the context ofPolynomial Optimization Problems, that consist in finding the infimum of a real polynomial ona real semialgebraic set defined by polynomial inequalities. We analyze the exactness of MomentMatrix (MoM) hierarchies, dual to the Sum of Squares (SoS) hierarchies, which are sequences ofconvex cones introduced by Lasserre to approximate measures and positive polynomials. Weinvestigate in particular flat truncation properties, which allow testing effectively when MoMexactness holds and recovering the minimizers.We show that the dual of the MoM hierarchy coincides with the SoS hierarchy extendedwith the real radical of the support of the defining quadratic module Q. We deduce thatflat truncation happens if and only if the support of the quadratic module associated withthe minimizers is of dimension zero. We also bound the order of the hierarchy at which flattruncation holds.As corollaries, we show that flat truncation and MoM exactness hold when regularityconditions, known as Boundary Hessian Conditions, hold (and thus that MoM exactness holdsgenerically); and when the support of the quadratic module Q is zero-dimensional. Effectivenumerical computations illustrate these flat truncation properties.

math.AC↗