Ronan Quarez
Piecewise Certificates of Positivity for matrix polynomials.
Mon, 01/11/2010 - 16:01 — SomNambulAuthors: Ronan Quarez
Subjects: Other
AbstractWe show that any symmetric positive definite homogeneous matrix polynomial
$M\in\R[x_1,...,x_n]^{m\times m}$ admits a piecewise semi-certificate, i.e. a
collection of identites $M(x)=\sum_jf_{i,j}(x)U_{i,j}(x)^TU_{i,j}(x)$ where
$U_{i,j}(x)$ is a matrix polynomial and $f_{i,j}(x)$ is a non negative
polynomial on a semi-algebraic subset $S_i$, where $\R^n=\cup_{i=1}^r S_i$.
This result generalizes to the setting of biforms.On positive Matrices which have a Positive Smith Normal Form.
Wed, 09/09/2009 - 22:08 — SomNambulAuthors: Ronan Quarez
Subjects: Rings and Algebras
AbstractIt is known that any symmetric matrix $M$ with entries in $\R[x]$ and which
is positive semi-definite for any substitution of $x\in\R$, has a Smith normal
form whose diagonal coefficients are constant sign polynomials in $\R[x]$. We
generalize this result by considering a symmetric matrix $M$ with entries in a
formally real principal domain $A$, we assume that $M$ is positive
semi-definite for any ordering on $A$ and, under one additionnal hypothesis
concerning non-real primes, we show that the Smith normal of $M$ is positive,
up to association.
User login
