Semialgebraic Subset Research Articles

Adjoints and canonical forms of tree amplituhedra

We consider semi-algebraic subsets of the Grassmannian of lines in three-space called tree amplituhedra. These arise in the study of scattering amplitudes from particle physics. Our main result states that tree amplituhedra in $\mathrm {Gr}(2,4)$ are positive geometries. The numerator of their canonical form plays the role of the adjoint in Wachspress geometry, and is uniquely determined by explicit interpolation conditions.

Rational certificates of non-negativity on semialgebraic subsets of cylinders

Let g1,…,gs∈R[X1,…,Xn,Y] and S={(x¯,y)∈Rn+1|g1(x¯,y)≥0,…,gs(x¯,y)≥0} be a non-empty, possibly unbounded, subset of a cylinder in Rn+1. Let f∈R[X1,…,Xn,Y] be a polynomial which is positive on S. We prove that, under certain additional assumptions, for any non-constant polynomial q∈R[Y] which is positive on R, there is a certificate of the non-negativity of f on S given by a rational function having as numerator a polynomial in the quadratic module generated by g1,…,gs and as denominator a power of q.

Approximation of singular semialgebraic sets by smooth ones with respect to the Hausdorff metric

We prove that every closed, nowhere dense semialgebraic subset of Rn of dimension k can be approximated with respect to the Hausdorff metric by Nash closed submanifolds; i.e. smooth semialgebraic closed submanifolds of Rn of the same dimension k. Moreover approximating Nash submanifolds can be provided by some semialgebraic family.

On polynomial images of a closed ball

In this work we approach the problem of determining which (compact) semialgebraic subsets of $\mathbb{R}^{n}$ are images under polynomial maps $f:\mathbb{R}^{m} \to \mathbb{R}^{n}$ of the closed unit ball $\overline{\mathcal{B}}_m$ centered at the origin of some Euclidean space $\mathbb{R}^{m}$ and that of estimating (when possible) which is the smallest $m$ with this property. Contrary to what happens with the images of $\mathbb{R}^{m}$ under polynomial maps, it is quite straightforward to provide basic examples of semialgebraic sets that are polynomial images of the closed unit ball. For instance, simplices, cylinders, hypercubes, elliptic, parabolic or hyperbolic segments (of dimension $n$) are polynomial images of the closed unit ball in $\mathbb{R}^{n}$. The previous examples (and other basic ones proposed in the article) provide a large family of ‘$n$-bricks’ and we find necessary and sufficient conditions to guarantee that a finite union of ‘$n$-bricks’ is again a polynomial image of the closed unit ball either of dimension $n$ or $n + 1$. In this direction, we prove: A finite union $\mathcal{S}$ of $n$-dimensional convex polyhedra is the image of the $n$-dimensional closed unit ball $\overline{\mathcal{B}}_{n}$ if and only if $\mathcal{S}$ is connected by analytic paths. The previous result can be generalized using the ‘$n$-bricks’ mentioned before and we show: If $\mathcal{S}_{1}, \ldots, \mathcal{S}_{\ell} \subset \mathbb{R}^n$ are ‘$n$-bricks’, the union $\mathcal{S} := \bigcup_{i=1}^{\ell} \mathcal{S}_i$ is the image of the closed unit ball $\overline{\mathcal{B}}_{n+1}$ of $\mathbb{R}^{n+1}$ under a polynomial map $f:\mathbb{R}^{n+1} \to \mathbb{R}^{n}$ if and only if $\mathcal{S}$ is connected by analytic paths.

Efficient simplicial replacement of semialgebraic sets

Abstract Designing an algorithm with a singly exponential complexity for computing semialgebraic triangulations of a given semialgebraic set has been a holy grail in algorithmic semialgebraic geometry. More precisely, given a description of a semialgebraic set $S \subset \mathbb {R}^k$ by a first-order quantifier-free formula in the language of the reals, the goal is to output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ , is semialgebraically homeomorphic to S. In this paper, we consider a weaker version of this question. We prove that for any $\ell \geq 0$ , there exists an algorithm which takes as input a description of a semialgebraic subset $S \subset \mathbb {R}^k$ given by a quantifier-free first-order formula $\phi $ in the language of the reals and produces as output a simplicial complex $\Delta $ , whose geometric realization, $|\Delta |$ is $\ell $ -equivalent to S. The complexity of our algorithm is bounded by $(sd)^{k^{O(\ell )}}$ , where s is the number of polynomials appearing in the formula $\phi $ , and d a bound on their degrees. For fixed $\ell $ , this bound is singly exponential in k. In particular, since $\ell $ -equivalence implies that the homotopy groups up to dimension $\ell $ of $|\Delta |$ are isomorphic to those of S, we obtain a reduction (having singly exponential complexity) of the problem of computing the first $\ell $ homotopy groups of S to the combinatorial problem of computing the first $\ell $ homotopy groups of a finite simplicial complex of size bounded by $(sd)^{k^{O(\ell )}}$ .

Geometry of Linear Convolutional Networks

We study the family of functions that are represented by a linear convolutional network (LCN). These functions form a semi-algebraic subset of the set of linear maps from input space to output space. In contrast, the families of functions represented by fully connected linear networks form algebraic sets. We observe that the functions represented by LCNs can be identified with polynomials that admit certain factorizations, and we use this perspective to describe the impact of the network's architecture on the geometry of the resulting function space. We further study the optimization of an objective function over an LCN, analyzing critical points in function space and in parameter space and describing dynamical invariants for gradient descent. Overall, our theory predicts that the optimized parameters of an LCN will often correspond to repeated filters across layers, or filters that can be decomposed as repeated filters. We also conduct numerical and symbolic experiments that illustrate our results and present an in-depth analysis of the landscape for small architectures.

An Algebraic Approach to Product-form Stationary Distributions for Some Reaction Networks

Exact results for product-form stationary distributions of Markov chains are of interest in different fields. In stochastic reaction networks (CRNs), stationary distributions are mostly known in special cases where they are of product-form. However, there is no full characterization of the classes of networks whose stationary distributions have product-form. We develop an algebraic approach to product-form stationary distributions in the framework of CRNs. Under certain hypotheses on linearity and decomposition of the state space for conservative CRNs, this gives sufficient and necessary algebraic conditions for product-form stationary distributions. Correspondingly, we obtain a semialgebraic subset of the parameter space that captures rates where, under the corresponding hypotheses, CRNs have product-form. We employ the developed theory to CRNs and some models of statistical mechanics, besides sketching the pertinence in other models from applied probability.

Catastrophe in Elastic Tensegrity Frameworks

We discuss elastic tensegrity frameworks made from rigid bars and elastic cables, depending on many parameters. For any fixed parameter values, the stable equilibrium position of the framework is determined by minimizing an energy function subject to algebraic constraints. As parameters smoothly change, it can happen that a stable equilibrium disappears. This loss of equilibrium is called catastrophe, since the framework will experience large-scale shape changes despite small changes of parameters. Using nonlinear algebra, we characterize a semialgebraic subset of the parameter space, the catastrophe set, which detects the merging of local extrema from this parametrized family of constrained optimization problems, and hence detects possible catastrophe. Tools from numerical nonlinear algebra allow reliable and efficient computation of all stable equilibrium positions as well as the catastrophe set itself.

GROTHENDIECK RINGS OF DEFINABLE SUBASSIGNMENTS AND EQUIVARIANT MOTIVIC MEASURES

This paper studies categories of definable subassignments with some category equivalences to semi-algebraic and constructible subsets of arc spaces of algebraic varieties. These equivalences lead to the identity of certain Grothendieck rings, which allows us to compare the motivic measure of Cluckers-Loeser with that of Denef-Loeser for certain classes of definable subassignments.

A p-adic variant of Kontsevich–Zagier integral operation rules and of Hrushovski–Kazhdan style motivic integration

Abstract We prove that if two semi-algebraic subsets of ℚ p n {\mathbb{Q}_{p}^{n}} have the same p-adic measure, then this equality can already be deduced using only some basic integral transformation rules. On the one hand, this can be considered as a positive answer to a p-adic analogue of a question asked by Kontsevich–Zagier in the reals (though the question in the reals is much harder). On the other hand, our result can also be considered as stating that over ℚ p {\mathbb{Q}_{p}} , universal motivic integration (in the sense of Hrushovski–Kazhdan) coincides with the usual p-adic integration.

The Stieltjes condition and multidimensional {mathcal {K}}-moment problems

We prove a solvability theorem for the Stieltjes problem on mathbb {R}^d which is based on the multivariate Stieltjes condition sum _{n=1}^infty L(x_j^{n})^{-1/(2n)} =+infty , j=1,dots ,d. This result is applied to derive a new solvability theorem for the moment problem on unbounded semi-algebraic subsets of mathbb {R}^d.

Detecting optimality and extracting solutions in polynomial optimization with the truncated GNS construction

A basic closed semialgebraic subset of {mathbb {R}}^{n} is defined by simultaneous polynomial inequalities p_{1}ge 0,ldots ,p_{m}ge 0. We consider Lasserre’s relaxation hierarchy to solve the problem of minimizing a polynomial over such a set. These relaxations give an increasing sequence of lower bounds of the infimum. In this paper we provide a new certificate for the optimal value of a Lasserre relaxation to be the optimal value of the polynomial optimization problem. This certificate is to check if a certain matrix has a generalized Hankel form. This certificate is more general than the already known certificate of an optimal solution being flat. In case we have detected optimality we will extract the potential minimizers with a truncated version of the Gelfand–Naimark–Segal construction on the optimal solution of the Lasserre relaxation. We prove also that the operators of this truncated construction commute if and only if the matrix of this modified optimal solution is a generalized Hankel matrix. This generalization of flatness will enable us to prove, with the use of the GNS truncated construction, a result of Curto and Fialkow on the existence of quadrature rule if the optimal solution is flat and a result of Xu and Mysovskikh on the existence of a Gaussian quadrature rule if the modified optimal solution is a generalized Hankel matrix . At the end, we provide a numerical linear algebraic algorithm for detecting optimality and extracting solutions of a polynomial optimization problem.

VC density of definable families over valued fields

We prove a tight bound on the number of realized 0=1 patterns (or equivalently on the Vapnik–Chervonenkis codensity) of definable families in models of the theory of algebraically closed valued fields with a non-archimedean valuation. Our result improves the best known result in this direction proved by Aschenbrenner, Dolich, Haskell, Macpherson and Starchenko, who proved a weaker bound in the restricted case where the characteristics of the field K and its residue field are both assumed to be 0. The bound obtained here is optimal and without any restriction on the characteristics. We obtain the aforementioned bound as a consequence of another result on bounding the Betti numbers of semi-algebraic subsets of certain Berkovich analytic spaces, mirroring similar results known already in the case of o-minimal structure and for real closed as well as algebraically closed fields. The latter result is the first result in this direction and is possibly of independent interest. Its proof relies heavily on recent results of Hrushovski and Loeser on the topology of semi-algebraic subsets of Berkovich analytic spaces.

Rational quartic spectrahedra

Rational quartic spectrahedra in $3$-space are semialgebraic convex subsets in $\mathbb{R} ^3$ of semidefinite, real symmetric $(4 \times 4)$-matrices, whose boundary admits a rational parameterization. The Zariski closure in $\mathbb{C}\mathbb{P} ^3$ of the boundary of a rational spectrahedron is a rational complex symmetroid. We give necessary conditions on the configurations of singularities of the corresponding real symmetroids in $\mathbb{R} \mathbb{P} ^3$ of rational quartic spectrahedra. We provide an almost exhaustive list of examples realizing the configurations, and conjecture that the missing example does not occur.

Gaussian graphical models with toric vanishing ideals

Gaussian graphical models are semi-algebraic subsets of the cone of positive definite covariance matrices. They are widely used throughout natural sciences, computational biology and many other fields. Computing the vanishing ideal of the model gives us an implicit description of the model. In this paper, we resolve two conjectures given by Sturmfels and Uhler. In particular, we characterize those graphs for which the vanishing ideal of the Gaussian graphical model is generated in degree 1 and 2. These turn out to be the Gaussian graphical models whose ideals are toric ideals, and the resulting graphs are the 1-clique sums of complete graphs.

Quantitative Singularity Theory for Random Polynomials

Abstract Motivated by Hilbert’s 16th problem we discuss the probabilities of topological features of a system of random homogeneous polynomials. The distribution for the polynomials is the Kostlan distribution. The topological features we consider are type-$W$ singular loci. This is a term that we introduce and that is defined by a list of equalities and inequalities on the derivatives of the polynomials. In technical terms a type-$W$ singular locus is the set of points where the jet of the Kostlan polynomials belongs to a semialgebraic subset $W$ of the jet space, which we require to be invariant under orthogonal change of variables. For instance, the zero set of polynomial functions or the set of critical points fall under this definition. We will show that, with overwhelming probability, the type-$W$ singular locus of a Kostlan polynomial is ambient isotopic to that of a polynomial of lower degree. As a crucial result, this implies that complicated topological configurations are rare. Our results extend earlier results from Diatta and Lerario who considered the special case of the zero set of a single polynomial. Furthermore, for a given polynomial function $p$ we provide a deterministic bound for the radius of the ball in the space of differentiable functions with center $p$, in which the $W$-singularity structure is constant.

Real Tropicalization and Analytification of Semialgebraic Sets

Abstract Let $K$ be a real closed field with a nontrivial non-archimedean absolute value. We study a refined version of the tropicalization map, which we call real tropicalization map, that takes into account the signs on $K$. We study images of semialgebraic subsets of $K^n$ under this map from a general point of view. For a semialgebraic set $S \subseteq K^n$ we define a space $S_r^{{\operatorname{an}}}$ called the real analytification, which we show to be homeomorphic to the inverse limit of all real tropicalizations of $S$. We prove a real analogue of the tropical fundamental theorem and show that the tropicalization of any semialgebraic set is described by tropicalization of finitely many inequalities, which are valid on the semialgebraic set. We also study the topological properties of real analytification and tropicalization. If $X$ is an algebraic variety, we show that $X_r^{{\operatorname{an}}}$ can be canonically embedded into the real spectrum $X_r$ of $X$, and we study its relation with the Berkovich analytification of $X$.

Numerical roadmap of smooth bounded real algebraic surface

For a smooth bounded real algebraic surface in Rn, a roadmap of it is a one-dimensional semi-algebraic subset of the surface whose intersection with each connected component of the surface is nonempty and semi-algebraically connected. In this paper, we introduce the notion of a numerical roadmap of a surface, which is a set of disjoint polygonal chains such that there is a bijective map between the chains and the connected components of a given roadmap of the surface. Moreover, the chains are ϵ-close to the connected components. We present an algorithm to compute such a numerical roadmap through constructing a topological graph. The topological graph also enables us to compute an approximate graph and a more intrinsic connectivity graph to represent the roadmap and its connectivity property, which is important for applications such as determining if two points on the surface belong to the same connected component and if so, finding a connected path between them.

Locally definable subgroups of semialgebraic groups

We prove the following instance of a conjecture stated in [P. E. Eleftheriou and Y. Peterzil, Definable quotients of locally definable groups, Selecta Math. (N.S.) 18(4) (2012) 885–903]. Let [Formula: see text] be an abelian semialgebraic group over a real closed field [Formula: see text] and let [Formula: see text] be a semialgebraic subset of [Formula: see text]. Then the group generated by [Formula: see text] contains a generic set and, if connected, it is divisible. More generally, the same result holds when [Formula: see text] is definable in any o-minimal expansion of [Formula: see text] which is elementarily equivalent to [Formula: see text]. We observe that the above statement is equivalent to saying: there exists an [Formula: see text] such that [Formula: see text] is an approximate subgroup of [Formula: see text].

Differentiable approximation of continuous semialgebraic maps

In this work we approach the problem of approximating uniformly continuous semialgebraic maps $f:S\to T$ from a compact semialgebraic set $S$ to an arbitrary semialgebraic set $T$ by semialgebraic maps $g:S\to T$ that are differentiable of class~${\mathcal C}^\nu$ for a fixed integer $\nu\geq1$. As the reader can expect, the difficulty arises mainly when one tries to keep the same target space after approximation. For $\nu=1$ we give a complete affirmative solution to the problem: such a uniform approximation is always possible. For $\nu \geq 2$ we obtain density results in the two following relevant situations: either $T$ is compact and locally ${\mathcal C}^\nu$ semialgebraically equivalent to a polyhedron, for instance when $T$ is a compact polyhedron; or $T$ is an open semialgebraic subset of a Nash set, for instance when $T$ is a Nash set. Our density results are based on a recent ${\mathcal C}^1$-triangulation theorem for semialgebraic sets due to Ohmoto and Shiota, and on new approximation techniques we develop in the present paper. Our results are sharp in a sense we specify by explicit examples.