Real Algebraic and Analytic Geometry |

A note on the representation of positive polynomials with structured sparsity.

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Submission: 2006, November 16.

*Abstract:
We consider real polynomials in finitely many variables. Let the variables
consist of finitely many blocks that are allowed to overlap in a certain way.
Let the solution set of a finite system of polynomial inequalities be
given where each inequality involves only variables of one block. We
investigate polynomials that are positive on such a set and sparse in
the sense that each monomial involves only variables of one block.
In particular, we derive a short and direct proof for Lasserre's theorem
of the existence of sums of squares certificates respecting the block
structure. The motivation for the results can be found in the literature
and stems from numerical methods using semidefinite programming to simulate or
control discrete-time behaviour of systems.*

Mathematics Subject Classification (2000): 11E25, 13J30, 14P10.

Keywords and Phrases: structured sparsity, sparse polynomial, positive polynomial, sum of squares.

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