Cordian Riener

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In this paper we study various approaches for exploiting symmetries in polynomial optimization problems within the framework of semidefinite 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 the(More)
Unlike the well known classical bounds due to Oleinik and Petrovskii, Thom and Milnor on the Betti numbers of (possibly non-symmetric) real algebraic varieties and semialgebraic sets, the above bound is polynomial in k when the degrees of the defining polynomials are bounded by a constant. Moreover, our bounds are asymptotically tight. As an application we(More)
A theorem of Voronoi asserts that a lattice is extreme if and only if it is perfect and eutactic. Very recently the classification of the perfect forms in dimension 8 has been completed [5]. There are 10916 perfect lattices. Using methods of linear programming, we are able to identify those that are additionally eutactic. In lower dimensions almost all(More)
Let d and k be positive integers. Let μ be a positive Borel measure on R2 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 degree at(More)
We give algorithms with polynomially bounded complexities (for fixed degrees) for computing the generalized Euler-Poincaré characteristic of semi-algebraic sets defined by symmetric polynomials. This is in contrast to the best complexity of the known algorithms for the same problem in the non-symmetric situation, which is singly exponential. This singly(More)
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 Ck and PC, defined by r polynomials of degrees at most d as a function of p, d and r. Unlike previous bounds these bounds are(More)
Let R be a real closed field. The problem of obtaining tight bounds on the Betti numbers of semi-algebraic subsets of R 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 Olĕınik and Petrovskĭı, Thom and Milnor. These bounds are all exponential in the number(More)
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 which acts(More)