arXiv subjects

Cordian Riener

Publications and source records attributed to Cordian Riener.

At least 37 records · Page 2Linked to original sources

arXiv2023

Optimization of trigonometric polynomials with crystallographic symmetry and spectral bounds for set avoiding graphs

Trigonometric polynomials are usually defined on the lattice of integers.We consider the larger class of weight and root lattices with crystallographic symmetry.This article gives a new approach to minimize trigonometric polynomials, which are invariant under the associated reflection group.The invariance assumption allows us to rewrite the objective function in terms of generalized Chebyshev polynomials. The new objective function is defined on a compact basic semi-algebraic set, so that we can benefit from the rich theory of polynomial optimization.We present an algorithm to compute the minimum: Based on the Hol-Scherer Positivstellensatz, we impose matrix-sums of squares conditions on the objective function in the Chebyshev basis.The degree of the sums of squares is weighted, defined by the root system. Increasing the degree yields a converging Lasserre-type hierarchy of lower bounds.This builds a bridge between trigonometric and polynomial optimization, allowing us to compare with existing techniques.The chromatic number of a set avoiding graph in the Euclidean space is defined through an optimal coloring.It can be computed via a spectral bound by minimizing a trigonometric polynomial. If the to be avoided set has crystallographic symmetry, our method has a natural application.Specifically, we compute spectral bounds for the first time for boundaries of symmetric polytopes.For several cases, the problem has such a simplified form that we can give analytical proofs for sharp spectral bounds.In other cases, we certify the sharpness numerically.

math.AG↗

arXiv2023

Linear slices of hyperbolic polynomials and positivity of symmetric polynomial functions

A real univariate polynomial of degree $n$ is called hyperbolic if all of its $n$ roots are on the real line. Such polynomials appear quite naturally in different applications, for example, in combinatorics and optimization. The focus of this article are families of hyperbolic polynomials which are determined through $k$ linear conditions on the coefficients. The coefficients corresponding to such a family of hyperbolic polynomials form a semi-algebraic set which we call a \emph{hyperbolic slice}. We initiate here the study of the geometry of these objects in more detail. The set of hyperbolic polynomials is naturally stratified with respect to the multiplicities of the real zeros and this stratification induces also a stratification on the hyperbolic slices. Our main focus here is on the \emph{local extreme points} of hyperbolic slices, i.e., the local extreme points of linear functionals, and we show that these correspond precisely to those hyperbolic polynomials in the hyperbolic slice which have at most $k$ distinct roots and we can show that generically the convex hull of such a family is a polyhedron. Building on these results, we give consequences of our results to the study of symmetric real varieties and symmetric semi-algebraic sets. Here, we show that sets defined by symmetric polynomials which can be expressed sparsely in terms of elementary symmetric polynomials can be sampled on points with few distinct coordinates. This in turn allows for algorithmic simplifications, for example, to verify that such polynomials are non-negative or that a semi-algebraic set defined by such polynomials is empty.

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arXiv2022

Equivariant algebraic and semi-algebraic geometry of infinite affine space

We study $\textrm{Sym}(\infty)$-orbit closures of not necessarily closed points in the Zariski spectrum of the infinite polynomial ring $\mathbb{C}[x_{ij}:\, i\in\mathbb{N},\,j\in[n]]$. Among others, we characterize invariant prime ideals in this ring. Furthermore, we study projections of basic equivariant semi-algebraic sets defined by $\textrm{Sym}(\infty)$ orbits of polynomials in $\mathbb{R}[x_{ij}:\, i\in\mathbb{N},\,j\in[n]]$. For $n=1$ we prove a quantifier elimination type result which fails for $n>1$.

math.AG↗

arXiv2021

Reflection groups and cones of sums of squares

We consider cones of real forms which are sums of squares forms and invariant by a (finite) reflection group. We show how the representation theory of these groups allows to use the symmetry inherent in these cones to give more efficient descriptions. We focus especially on the $A_{n}$, $B_n$, and $D_n$ case where we use so called higher Specht polynomials to give a uniform description of these cones. These descriptions allow us, for example, to study the connection of these cones to non-negative forms. In particular, we give a new proof of a result by Harris who showed that every non-negative ternary even symmetric octic form is a sum of squares.

math.AG↗

arXiv2021

Symmetry reduction in AM/GM-based optimization

The arithmetic mean/geometric mean-inequality (AM/GM-inequality) facilitates classes of non-negativity certificates and of relaxation techniques for polynomials and, more generally, for exponential sums. Here, we present a first systematic study of the AM/GM-based techniques in the presence of symmetries under the linear action of a finite group. We prove a symmetry-adapted representation theorem and develop techniques to reduce the size of the resulting relative entropy programs. We study in more detail the complexity gain in the case of the symmetric group. In this setup, we can show in particular certain stabilization results. We exhibit several sequences of examples in growing dimensions where the size of the problem stabilizes. Finally, we provide some numerical results, emphasizing the computational speed-up.

math.OC↗

arXiv2021

Return of the plane evolute

Below we consider the evolutes of plane real-algebraic curves and discuss some of their complex and real-algebraic properties. In particular, for a given degree $d\ge 2$, we provide lower bounds for the following four numerical invariants: 1) the maximal number of times a real line can intersect the evolute of a real-algebraic curve of degree $d$; 2) the maximal number of real cusps which can occur on the evolute of a real-algebraic curve of degree $d$; 3) the maximal number of (cru)nodes which can occur on the dual curve to the evolute of a real-algebraic curve of degree $d$; 4) the maximal number of (cru)nodes which can occur on the evolute of a real-algebraic curve of degree $d$.

math.AG↗

arXiv2021

Vandermonde varieties, mirrored spaces, and the cohomology of symmetric semi-algebraic sets

Let $\mathrm{R}$ be a real closed field. We prove that for each fixed $\ell, d \geq 0$, there exists an algorithm that takes as input a quantifier-free first order formula $Φ$ with atoms $P=0, P > 0, P < 0 \text{ with } P \in \mathcal{P} \subset \mathrm{D}[X_1,\ldots,X_k]^{\mathfrak{S}_k}_{\leq d}$, where $\mathrm{D}$ is an ordered domain contained in $\mathrm{R}$, and computes the ranks of the first $(\ell+1)$ cohomology groups, of the symmetric semi-algebraic set defined by $Φ$. The complexity of this algorithm (measured by the number of arithmetic operations in $\mathrm{D}$) is bounded by a \emph{polynomial} in $k$ and $\mathrm{card}(\mathcal{P})$ (for fixed $d$ and $\ell$). This result contrasts with the $\mathbf{PSPACE}$-hardness of the problem of computing just the zero-th Betti number (i.e. the number of semi-algebraically connected components) in the general case for $d \geq 2$ (taking the ordered domain $\mathrm{D}$ to be equal to $\mathbb{Z}$). The above algorithmic result is built on new representation theoretic results on the cohomology of symmetric semi-algebraic sets. We prove that the Specht modules corresponding to partitions having long lengths cannot occur with positive multiplicity in the isotypic decompositions of low dimensional cohomology modules of closed semi-algebraic sets defined by symmetric polynomials having small degrees. This result generalizes prior results obtained by the authors giving restrictions on such partitions in terms of their ranks, and is the key technical tool in the design of the algorithm mentioned in the previous paragraph.

math.AG↗

arXiv2021

Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations

An ideal of polynomials is symmetric if it is closed under permutations of variables. We relate general symmetric ideals to the so called Specht ideals generated by all Specht polynomials of a given shape. We show a connection between the leading monomials of polynomials in the ideal and the Specht polynomials contained in the ideal. This provides applications in several contexts. Most notably, this connection gives information about the solutions of the corresponding set of equations. From another perspective, it restricts the isotypic decomposition of the ideal viewed as a representation of the symmetric group.

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arXiv2020

Symmetric nonnegative forms and sums of squares

We study symmetric nonnegative forms and their relationship with symmetric sums of squares. For a fixed number of variables $n$ and degree $2d$, symmetric nonnegative forms and symmetric sums of squares form closed, convex cones in the vector space of $n$-variate symmetric forms of degree $2d$. Using representation theory of the symmetric group we characterize both cones in a uniform way. Further, we investigate the asymptotic behavior when the degree $2d$ is fixed and the number of variables $n$ grows. Here, we show that, in sharp contrast to the general case, the difference between symmetric nonnegative forms and sums of squares does not grow arbitrarily large for any fixed degree $2d$. We consider the case of symmetric quartic forms in more detail and give a complete characterization of quartic symmetric sums of squares. Furthermore, we show that in degree $4$ the cones of nonnegative symmetric forms and symmetric sums of squares approach the same limit, thus these two cones asymptotically become closer as the number of variables grows. We conjecture that this is true in arbitrary degree $2d$.

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arXiv2018

Real root finding for equivariant semi-algebraic systems

Let $R$ be a real closed field. We consider basic semi-algebraic sets defined by $n$-variate equations/inequalities of $s$ symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by $2d < n$. Such a semi-algebraic set is invariant by the action of the symmetric group. We show that such a set is either empty or it contains a point with at most $2d-1$ distinct coordinates. Combining this geometric result with efficient algorithms for real root finding (based on the critical point method), one can decide the emptiness of basic semi-algebraic sets defined by $s$ polynomials of degree $d$ in time $(sn)^{O(d)}$. This improves the state-of-the-art which is exponential in $n$. When the variables $x_1, \ldots, x_n$ are quantified and the coefficients of the input system depend on parameters $y_1, \ldots, y_t$, one also demonstrates that the corresponding one-block quantifier elimination problem can be solved in time $(sn)^{O(dt)}$.

cs.SC↗

arXiv2018

Generalized eigenvalue methods for Gaussian quadrature rules

A quadrature rule of a measure $μ$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $μ$ for all polynomials up to some fixed degree. In this paper, we present a bivariate polynomial whose roots parametrize the nodes of minimal quadrature rules for measures on the real line. We give two symmetric determinantal formulas for this polynomial, which translate the problem of finding the nodes to solving a generalized eigenvalue problem.

math.NA↗

arXiv2018

On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms

Let $\mathrm{R}$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by polynomials of degrees bounded by $d$ vanishes in dimensions $d$ and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of $d^{O(d)} s^d k^{\lfloor d/2 \rfloor-1} $ on the equivariant Betti numbers of closed symmetric semi-algebraic subsets of $\mathrm{R}^k$ defined by quantifier-free formulas involving $s$ symmetric polynomials of degrees bounded by $d$, where $1 < d \ll s,k$. This bound is tight up to a factor depending only on $d$. These results significantly improve upon those obtained previously which were proved using different techniques. Our new methods are quite general, and also yield bounds on the equivariant Betti numbers of certain special classes of symmetric definable sets (definable sets symmetrized by pulling back under symmetric polynomial maps of fixed degree) in arbitrary o-minimal structures over $\mathrm{R}$. Finally, we utilize our new approach to obtain an algorithm with polynomially bounded complexity for computing these equivariant Betti numbers. In contrast, the problem of computing the ordinary Betti numbers of (not necessarily symmetric) semi-algebraic sets is considered to be an intractable problem, and all known algorithms for this problem have doubly exponential complexity.

math.AT↗

arXiv2017

Optimization approaches to quadrature: new characterizations of Gaussian quadrature on the line and quadrature with few nodes on plane algebraic curves, on the plane and in higher dimensions

Let $d$ and $k$ be positive integers. Let $μ$ be a positive Borel measure on $\mathbb{R}^2$ possessing finite moments up to degree $2d-1$. If the support of $μ$ is contained in an algebraic curve of degree $k$, then we show that there exists a quadrature rule for $μ$ with at most $dk$ many nodes all placed on the curve (and positive weights) that is exact on all polynomials of degree at most $2d-1$. This generalizes both Gauss and (the odd degree case of) Szegő quadrature where the curve is a line and a circle, respectively, to arbitrary plane algebraic curves. We use this result to show that, without any hypothesis on the support of $μ$, there is always a cubature rule for $μ$ with at most $\frac32d(d-1)+1$ many nodes. In both results, we show that the quadrature or cubature rule can be chosen such that its value on a certain positive definite form of degree $2d$ is minimized. We characterize the unique Gaussian quadrature rule on the line as the one that minimizes this value or several other values as for example the sum of the nodes' distances to the origin. The tools we develop should prove useful for obtaining similar results in higher-dimensional cases although at the present stage we can present only partial results in that direction.

math.NA↗

arXiv2017

On irreducible components of real exponential hypersurfaces

Fix any algebraic extension $\mathbb K$ of the field $\mathbb Q$ of rationals. In this article we study exponential sets $V\subset \mathbb R^n$. Such sets are described by the vanishing of so called exponential polynomials, i.e., polynomials with coefficients from $\mathbb K$, in $n$ variables, and in $n$ exponential functions. The complements of all exponential sets in $\mathbb R^n$ form a Noethrian topology on $\mathbb R^n$, which we will call Zariski topology. Let $P \in {\mathbb K}[X_1, \ldots ,X_n,U_1, \ldots ,U_n]$ be a polynomial such that $$V=\{ \mathbf{x}=(x_1, \ldots , x_n) \in \mathbb R^n| P(\mathbf{x}, e^{x_1}, \ldots ,e^{x_n})=0 \}.$$ The main result of this paper states that, under Schanuel's conjecture over the reals, an exponential set $V$ of codimension 1, for which the real algebraic set $\rm Zer(P)$ is irreducible over $\mathbb K$, either is irreducible (with respect to the Zariski topology) or every of its irreducible components of codimension 1 is a rational hyperplane through the origin. The family of all possible hyperplanes is determined by monomials of $P$. In the case of a single exponential (i.e., when $P$ is independent of $U_2, \ldots , U_n$) stronger statements are shown which are independent of Schanuel's conjecture.

math.AG↗

arXiv2017

Efficient algorithms for computing the Euler-Poincaré characteristic of symmetric semi-algebraic sets

Let $\mathrm{R}$ be a real closed field and $\mathrm{D} \subset \mathrm{R}$ an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincaré characteristic of real algebraic as well as semi-algebraic subsets of $\mathrm{R}^k$, which are defined by symmetric polynomials with coefficients in $\mathrm{D}$. We give algorithms for computing the generalized Euler-Poincaré characteristic of such sets, whose complexities measured by the number the number of arithmetic operations in $\mathrm{D}$, are polynomially bounded in terms of $k$ and the number of polynomials in the input, assuming that the degrees of the input polynomials are bounded by a constant. This is in contrast to the best complexity of the known algorithms for the same problems in the non-symmetric situation, which are singly exponential. This singly exponential complexity for the latter problem is unlikely to be improved because of hardness result ($\#\mathbf{P}$-hardness) coming from discrete complexity theory.

math.AG↗

arXiv2017

Reflection groups, reflection arrangements, and invariant real varieties

Let $X$ be a nonempty real variety that is invariant under the action of a reflection group $G$. We conjecture that if $X$ is defined in terms of the first $k$ basic invariants of $G$ (ordered by degree), then $X$ meets a $k$-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most $3$, and $F_4$ and we give computational evidence for $H_4$. This is a generalization of Timofte's degree principle to reflection groups. For general reflection groups, we compute nontrivial upper bounds on the minimal dimension of flats of the reflection arrangement meeting $X$ from the combinatorics of parabolic subgroups. We also give generalizations to real varieties invariant under Lie groups.

math.AG↗

arXiv2017

On the isotypic decomposition of cohomology modules of symmetric semi-algebraic sets: polynomial bounds on multiplicities

We consider symmetric (under the action of products of finite symmetric groups) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of degrees bounded by a fixed constant $d$. We prove that if a Specht module, $\mathbb{S}^λ$, appears with positive multiplicity in the isotypic decomposition of the cohomology modules of such sets, then the rank of the partition $λ$ is bounded by $O(d)$. This implies a polynomial (in the dimension of the ambient space) bound on the number of such modules. Furthermore, we prove a polynomial bound on the multiplicities of those that do appear with positive multiplicity in the isotypic decomposition of the above mentioned cohomology modules. We give some applications of our methods in proving lower bounds on the degrees of defining polynomials of certain symmetric semi-algebraic sets, as well as improved bounds on the Betti numbers of the images under projections of (not necessarily symmetric) bounded real algebraic sets, improving in certain situations prior results of Gabrielov, Vorobjov and Zell.

math.AG↗

arXiv2016

Bounding the equivariant Betti numbers of symmetric semi-algebraic sets

Let $\mathrm{R}$ be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of $\mathrm{R}^k$ in terms of the number and degrees of the defining polynomials has been an important problem in real algebraic geometry with the first results due to Ole{\uı}nik and Petrovski{\uı}, Thom and Milnor. These bounds are all exponential in the number of variables $k$. Motivated by several applications in real algebraic geometry, as well as in theoretical computer science, where such bounds have found applications, we consider in this paper the problem of bounding the equivariant Betti numbers of symmetric algebraic and semi-algebraic subsets of $\mathrm{R}^k$. We obtain several asymptotically tight upper bounds. In particular, we prove that if $S\subset \mathrm{R}^k$ is a semi-algebraic subset defined by a finite set of $s$ symmetric polynomials of degree at most $d$, then the sum of the $\mathfrak{S}_k$-equivariant Betti numbers of $S$ with coefficients in $\mathbb{Q}$ is bounded by $(skd)^{O(d)}$. Unlike the classical bounds on the ordinary Betti numbers of real algebraic varieties and semi-algebraic sets, the above bound is polynomial in $k$ when the degrees of the defining polynomials are bounded by a constant. As an application we improve the best known bound on the ordinary Betti numbers of the projection of a compact algebraic set improving for any fixed degree the best previously known bound for this problem due to Gabrielov, Vorobjov and Zell.

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