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28 March 2024
 
  » arxiv » math.AG/0412400

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S.o.s. approximation of polynomials nonnegative on a real algebraic set
Jean B. Lasserre ;
Date 20 Dec 2004
Subject Algebraic Geometry MSC-class: 11E25; 12D15; 13P05; 12Y05; 90C22; 90C25 | math.AG
AbstractWith every real polynomial $f$, we associate a family ${f_{epsilon r}}_{epsilon, r}$ of real polynomials, in explicit form in terms of $f$ and the parameters $epsilon>0,rin N$, and such that $Vert f-f_{epsilon r}Vert_1 o 0$ as $epsilon o 0$. Let $Vsubset R^n$ be a real algebraic set described by finitely many polynomials equations $g_j(x)=0,jin J$, and let $f$ be a real polynomial, nonnegative on $V$. We show that for every $epsilon>0$, there exist nonnegative scalars ${lambda_j(epsilon)}_{jin J}$ such that, for all $r$ sufficiently large, $$f_{epsilon r}+sum_{jin J} lambda_j(epsilon) g_j^2,quad is a sum of squares.$$ This representation is an obvious certificate of nonnegativity of $f_{epsilon r}$ on $V$, and very specific in terms of the $g_j$ that define the set $V$. In particular, it is valid with {it no} assumption on $V$. In addition, this representation is also useful from a computation point of view, as we can define semidefinite programing relaxations to approximate the global minimum of $f$ on a real algebraic set $V$, or a semi-algebraic set $K$, and again, with {it no} assumption on $V$ or $K$.
Source arXiv, math.AG/0412400
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