Blar i forfatter Artikler, rapporter og annet (matematikk og statistikk) "Riener, Cordian"
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Generalized eigenvalue methods for Gaussian quadrature rules
Blekherman, Grigoriy; Kummer, Mario; Riener, Cordian; Schweighofer, Markus; Vinzant, Cynthia (Journal article; Tidsskriftartikkel; Peer reviewed, 2020)A quadrature rule of a measure <i>µ</i> on the real line represents a conic combination of finitely many evaluations at points, called nodes, that agrees with integration against <i>µ</i> 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 ... -
On the equivariant Betti numbers of symmetric definable sets: vanishing, bounds and algorithms
Basu, Saugata; Riener, Cordian (Journal article; Tidsskriftartikkel; Peer reviewed, 2018-03-02)Let R be a real closed field. We prove that for any fixed <i>d</i>, the equivariant rational cohomology groups of closed symmetric semi-algebraic subsets of R<sup><i>k</i></sup> defined by polynomials of degrees bounded by <i>d</i> vanishes in dimensions <i>d</i> and larger. This vanishing result is tight. Using a new geometric approach we also prove an upper bound of [mathematical formula] on the ... -
On the Isotypic Decomposition of Cohomology Modules of Symmetric Semi-algebraic Sets: Polynomial Bounds on Multiplicities
Basu, Saugata; Riener, Cordian (Journal article; Tidsskriftartikkel; Peer reviewed, 2018-04-30)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 <i>d</i>. We prove that if a Specht module, S<sup>λ</sup>, appears with positive multiplicity in the isotypic decomposition of the ... -
Real Root Finding for Equivariant Semi-algebraic Systems
Riener, Cordian; Safey el Din, Mohab (Conference object; Konferansebidrag, 2018)Let <i><b>R</b></i> be a real closed field. We consider basic semi-algebraic sets defined by <i>n</i>-variate equations/inequalities of s symmetric polynomials and an equivariant family of polynomials, all of them of degree bounded by 2<i>d</i> < <i>n</i>. 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 ... -
Reflection groups, arrangements, and invariant real varieties
Friedl, Tobias; Riener, Cordian; Sanyal, Raman (Journal article; Tidsskriftartikkel; Peer reviewed, 2017-10-18)Let <i>X</i> be a nonempty real variety that is invariant under the action of a reflection group <i>G</i>. We conjecture that if <i>X</i> is defined in terms of the first <i>k</i> basic invariants of <i>G</i> (ordered by degree), then <i>X</i> meets a <i>k</i>-dimensional flat of the associated reflection arrangement. We prove this conjecture for the infinite types, reflection groups of rank at most ... -
Symmetric ideals, Specht polynomials and solutions to symmetric systems of equations
Moustrou, Philippe; Riener, Cordian; Verdure, Hugues (Journal article; Tidsskriftartikkel; Peer reviewed, 2021-02-18)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 ... -
Symmetric Non-Negative Forms and Sums of Squares
Blekherman, Grigoriy; Riener, Cordian (Journal article; Tidsskriftartikkel; Peer reviewed, 2020-05-21)We study symmetric non-negative forms and their relationship with symmetric sums of squares. For a fixed number of variables <i>n</i> and degree 2<i>d</i>, symmetric non-negative forms and symmetric sums of squares form closed, convex cones in the vector space of <i>n</i>-variate symmetric forms of degree 2<i>d</i>. Using representation theory of the symmetric group we characterize both cones in a ... -
Symmetry Reduction in AM/GM-Based Optimization
Verdure, Hugues; Moustrou, Philippe; Naumann, Helen; Riener, Cordian; Theobald, Thorsten (Journal article; Tidsskriftartikkel; Peer reviewed, 2022)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 prove a symmetry-adapted representation theorem and ...