arXiv subjects

Marcelo Valdettaro

Publications and source records attributed to Marcelo Valdettaro.

4 recordsLinked to original sources

arXiv2023

Strictly positive polynomials in the boundary of the SOS cone

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which corresponds to polynomials that vanish at at least one point. We focus on the part of the boundary which is not shared, corresponding to strictly positive polynomials. For the cases of polynomials of degree 6 in 3 variables and degree 4 in 4 variables, this boundary has been completely characterized by G. Blekherman. For the cases of more variables or higher degree, results by G. Blekherman, R. Sinn and M. Velasco and other authors based on general conjectures give bounds for the maximum number of polynomials that can appear in a SOS decomposition and the maximum rank of the matrices in the Gram spectrahedron. Combining theoretical results and computational techniques, we compute examples that allow us to prove the optimality of the bounds for all degrees and number of variables. Additionally, we give examples for the following problems: examples in the boundary of the cone that are the sum of less than $n$ squares and have common complex roots, and examples of polynomials in the boundary with length larger than the expected from the dimension.

math.AG↗

arXiv2018

Closed formula for univariate subresultants in multiple roots

We generalize Sylvester single sums to multisets (sets with repeated elements), and show that these sums compute subresultants of two univariate polyomials as a function of their roots independently of their multiplicity structure. This is the first closed formula for subresultants in terms of roots that works for arbitrary polynomials, previous efforts only handled special cases. Our extension involves in some cases confluent Schur polynomials, and is obtained by using multivariate symmetric interpolation via an Exchange Lemma.

math.AC↗

arXiv2017

Subresultants in multiple roots: an extremal case

We provide explicit formulae for the coefficients of the order-d polynomial subresultant of (x-α)^m and (x-β)^n with respect to the set of Bernstein polynomials \{(x-α)^j(x-β)^{d-j}, \, 0\le j\le d\}. They are given by hypergeometric expressions arising from determinants of binomial Hankel matrices.

math.AC↗

arXiv2016

Symmetric Interpolation, Exchange Lemma and Sylvester Sums

The theory of symmetric multivariate Lagrange interpolation is a beautiful but rather unknown tool that has many applications. Here we derive from it an Exchange Lemma that allows to explain in a simple and natural way the full description of the double sum expressions introduced by Sylvester in 1853 in terms of subresultants and their Bezout coefficients.

math.AC↗