Cordian Riener

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• Dr phil. nat.
• Professor (Full) at UiT The Arctic University of Norway

Let d and k be positive integers. Let µ be a positive Borel measure on R 2 possessing 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 de...

In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semi definite programming relaxations. Our special focus is on constrained problems especially when the symmetric group is acting on the variables. In particular, we investigate the concept of block decomposition within th...

Let $\mathbf{R}$ be a real closed field. We prove upper bounds on the equivariant Betti numbers of symmetric algebraic and semi-algebraic subsets of $\mathbf{R}^k$. More precisely, we prove that if $S\subset \mathbf{R}^k$ is a semi-algebraic subset defined by a finite set of $s$ symmetric polynomials of degree at most $d$, then the sum of the $\mat...

We consider symmetric (as well as multi-symmetric) real algebraic varieties and semi-algebraic sets, as well as symmetric complex varieties in affine and projective spaces, defined by polynomials of fixed degrees. We give polynomial (in the dimension of the ambient space) bounds on the number of irreducible representations of the symmetric group wh...

Let $\R$ be a real closed field. We prove that for any fixed $d$, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $\R^k$ 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^{...

A semi-algebraic set is a subset of $\mathbb{R}^n$ defined by a finite collection of polynomial equations and inequalities. In this paper, we investigate the problem of determining whether two points in such a set belong to the same connected component. We focus on the case where the defining equations and inequalities are invariant under the natur...

In this presentation we discuss four natural real algebraic problems related to the study of evolutes of real plane algebraic curves

We provide a new approach to the optimization of trigonometric polynomials with crystallographic symmetry. This approach widens the bridge between trigonometric and polynomial optimization. The trigonometric polynomials considered are supported on weight lattices associated to crystallographic root systems and are assumed invariant under the associ...

We study nonnegative and sums of squares symmetric (and even symmetric) functions of fixed degree. We can think of these as limit cones of symmetric nonnegative polynomials and symmetric sums of squares of fixed degree as the number of variables goes to infinity. We compare these cones, including finding explicit examples of nonnegative polynomials...

Let $G$ be a finite group acting linearly on $\mathbb{R}^n$. A celebrated Theorem of Procesi and Schwarz gives an explicit description of the orbit space $\mathbb{R}^n /\!/G$ as a basic closed semi-algebraic set. We give a new proof of this statement and another description as a basic closed semi-algebraic set using elementary tools from real algeb...

This chapter investigates how symmetries can be used to reduce the computational complexity in polynomial optimization problems. A focus will be specifically given on the Moment-SOS hierarchy in polynomial optimization, where results from representation theory and invariant theory of groups can be used. In addition, symmetry reduction techniques wh...

Polynomial optimization is a fascinating field of study that has revolutionized the way we approach nonlinear problems described by polynomial constraints. The applications of this field range from production planning processes to transportation, energy consumption, and resource control. This introductory book explores the latest research developm...

In this paper we present a new semidefinite programming hierarchy for covering problems in compact metric spaces. Over the last years, these kind of hierarchies were developed primarily for geometric packing and for energy minimization problems; they frequently provide the best known bounds. Starting from a semidefinite programming hierarchy for th...

The classes of sums of arithmetic-geometric exponentials (SAGE) and of sums of nonnegative circuit polynomials (SONC) provide nonnegativity certificates which are based on the inequality of the arithmetic and geometric means. We study the cones of symmetric SAGE and SONC forms and their relations to the underlying symmetric nonnegative cone. As mai...

In this paper, we consider the problem of deciding the existence of real solutions to a system of polynomial equations having real coefficients, and which are invariant under the action of the symmetric group. We construct and analyze a Monte Carlo probabilistic algorithm which solves this problem, under some regularity assumptions on the input, by...

This chapter investigates how symmetries can be used to reduce the computational complexity in polynomial optimization problems. A focus will be specifically given on the Moment-SOS hierarchy in polynomial optimization, where results from representation theory and invariant theory of groups can be used. In addition, symmetry reduction techniques wh...

We study the geometry of the image of the nonnegative orthant under the power-sum map and the elementary symmetric polynomials map. After analyzing the image in finitely many variables, we concentrate on the limit as the number of variables approaches infinity. We explain how the geometry of the limit plays a crucial role in undecidability results...

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 objecti...

The study of symmetry as a structural property of algebraic objects is one of the fundamental pillows of the developments of modern mathematics, most prominently beginning with the work of Abel and Galois. The focus of the workshop was on permutation actions of the symmetric group on polynomial rings and algebraic and semi-algebraic sets. More conc...

We present the results of our recent article [4] and discuss its applications [5]. A finite group with an integer representation has a multiplicative action on the ring of Laurent polynomials, which is induced by a nonlinear action on the compact torus. We study the structure of the orbit space as the image of the fundamental invariants. For the We...

Specht polynomials classically realize the irreducible representations of the symmetric group. The ideals defined by these polynomials provide a strong connection with the combinatorics of Young tableaux and have been intensively studied by several authors. We initiate similar investigations for the ideals defined by the Specht polynomials associat...

The arithmetic mean/geometric mean inequality (AM/GM inequality) facilitates classes of nonnegativity 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...

A finite group with an integer representation has a multiplicative action on the ring of Laurent polynomials, which is induced by a nonlinear action on the complex torus. We study the structure of the associated orbit space as the image of the fundamental invariants. For the Weyl groups of types A, B, C and D, this image is a compact basic semi--al...

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{Sy...

A real univariate polynomial of degree $n$ is called hyperbolic if all of its $n$ roots are on the real line. We study families of hyperbolic polynomials defined by $k$ linear conditions on the coefficients. We show that the polynomials corresponding to local extreme points of such families have at most $k$ distinct roots. Furthermore, we find that...

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 $d$;...

Let R be a real closed field. We prove that for each fixed ℓ,d≥0, there exists an algorithm that takes as input a quantifier-free first-order formula Φ with atoms P=0,P>0,P<0withP∈P⊂D[X1,…,Xk]≤dSk, where D is an ordered domain contained in R, and computes the ranks of the first ℓ+1 cohomology groups, of the symmetric semi-algebraic set defined by Φ...

We study the relationship between symmetric nonnegative forms and symmetric sums of squares. Our particular emphasis is on the asymptotic behavior when the degree 2d is fixed and the number of variables $n$ grows. We show that in sharp contrast to the general case the difference between symmetric forms and sums of squares does not grow arbitrarily...

#P-hardness of computing matrix immanants are proved for each member of a broad class of shapes and restricted sets of matrices. The class is characterized in the following way. If a shape of size $n$ in it is in form $(w,\mathbf{1}+\lambda)$ or its conjugate is in that form, where $\mathbf{1}$ is the all-$1$ vector, then $|\lambda|$ is $n^{\vareps...

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. W...

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...

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 pol...

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...

Let $\mathrm{R}$ be a real closed field, $d,k \in \mathbb{Z}_{> 0}$, $\mathbf{y} =(y_1,\ldots,y_d) \in \mathrm{R}^d$, and let $V_{d,\mathbf{y}}^{(k)}$ denote the Vandermonde variety defined by $p_1^{(k)} = y_1, \ldots, p_d^{(k)} = y_d$, where $p_j^{(k)} = \sum_{i=1}^{k} X_i^j$. Then, the cohomology groups $\mathrm{H}^*(V_{d,\mathbf{y}}^{(k)},\mathb...

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 O(d). This improves the state-of-the-art which is exponential in n . When the variables x1, łdots, xn are quantified and the coeffic...

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...

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...

A quadrature rule of a measure $\mu$ on the real line represents a convex combination of finitely many evaluations at points, called nodes, that agrees with integration against $\mu$ 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 measure...

We give algebraic conditions under which a real variety (or semialgebraic set) invariant under a reflection group meets low-dimensional strata of the corresponding reflection arrangement. This generalizes Timofte's (half-)degree principle and Riener's J-sparsity to general reflection groups. Our methods use basic insights into the semialgebraic geo...

Fix any algebraic extension $K$ of the field $Q$ of rationals. In this article we study {\em exponential sets} $V\subset R^n$. Such sets are described by the vanishing of so called {\em exponential polynomials}, i.e., polynomials with coefficients from $K$, in $n$ variables, and in $n$ exponential functions. The complements of all exponential sets...

Let R be a real closed field and D ⊂ R an ordered domain. We consider the algorithmic problem of computing the generalized Euler-Poincarécharacteristic of real algebraic as well as semi-algebraic subsets of Rk, which are defined by symmetric polynomials with coefficients in D. We give algorithms for computing the generalized Euler-Poincaré characte...

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 uppe...

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 a...

Let $R$ be a real closed field. We prove that for any fixed $$4, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of $R^k$ 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...

Let $d$ and $k$ be positive integers. Let $\mu$ be a positive Borel measure on $\mathbb{R}^2$ possessing finite moments up to degree $2d-1$. If the support of $\mu$ is contained in an algebraic curve of degree $k$, then we show that there exists a quadrature rule for $\mu$ with at most $dk$ many nodes all placed on the curve (and positive weights)...

We prove graded bounds on the individual Betti numbers of affine and projective complex varieties. In particular, we give for each $p,d,r$, explicit bounds on the $p$-th Betti numbers of affine and projective subvarieties of $\mathrm{C}^k$, $\mathbb{P}^k_{\mathrm{C}}$, as well as products of projective spaces, defined by $r$ polynomials of degrees...

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 group...

The question how to certify non-negativity of a polynomial function lies at the heart of Real Algebra and also has important applications to Optimization. In this article we investigate the question of non-negativity in the context of multisymmetric polynomials. In this setting we generalize the characterization of non-negative symmetric polynomial...

The question of how to certify the non-negativity of a polynomial function lies at the heart of Real Algebra and it also has important applications to Optimization. In the setting of symmetric polynomials Timofte provided a useful way of certifying non-negativity of symmetric polynomials that are of a fixed degree. In this note we present more gene...

In this note we aim to give a new, elementary proof of a statement that was first proved by Timofte. It says that a symmetric real polynomial $F$ of degree $d$ in $n$ variables is positive on $\R^n$ (on $\R^{n}_{\geq 0}$) if and only if it is so on the subset of points with at most $\max\{\lfloor d/2\rfloor,2\}$ distinct components. We deduce Timof...

This note provides a new approach to a result of Foregger [T.H. Foregger, On the relative extrema of a linear combination of elementary symmetric functions, Linear Multilinear Algebra 20 (1987) pp. 377-385] and related earlier results by Keilson [J. Keilson, A theorem on optimum allocation for a class of symmetric multilinear return functions, J. M...

A theorem of Voronoi asserts that a lattice is extreme if and only if it is perfect and eutactic. Very recently the classi- fication of the perfect forms in dimension 8 has been completed. There are 10916 perfect lattices. Using methods of linear programming, we are able to identify those that are additionally eutactic. In lower dimensions almost a...

This is a short survey (in German) of relatively recent (1998--2008) research activities at the border between optimization, numerical linear algebra, functional analysis and real algebraic geometry. The focus in on the use of semidefinite programming (linear programming in the convex cone of symmetric matrices with non-negative eigenvalues, see Se...