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What and Who
Title:Effectivity Issues and Results for Hilbert's 17th Problem
Speaker:Marie-Francoise Roy
coming from:Universit├ę de Rennes 1
Speakers Bio:
Event Type:Talk
Visibility:D1, D2, D3, D4, D5, RG1, SWS, MMCI
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Level:MPI Audience
Language:English
Date, Time and Location
Date:Thursday, 25 February 2016
Time:16:15
Duration:60 Minutes
Location:Saarbr├╝cken
Building:E1 4
Room:024
Abstract
If a polynomial is everywhere non negative, it is a sum of squares of rational fractions (which is the positive solution of Hilbert's 17th problem). This is an example of a certificate for positivity (more precisely non-negativity), i.e. an algebraic identify certifying that the polynomial is non-negative.

But how to construct this sum of squares from a proof of the non negativity?

Artin's initial proof of Hilbert's 17th problem in 1927 was very indirect, and did not produce explicitly the sum of squares. Effective methods have been designed since then (starting from Kreisel and his students) but the degree estimate of the corresponding constructions were primitive recursive functions.

In this talk, we recall the initial proof of Artin and present a new constructive proof for Hilbert's 17th problem, and show that following this construction, the degree of the polynomials in the identity is bounded by an elementary recursive function in the number of variables, and the degree of the polynomial (more precisely a tower of five exponentials). We end with a discussion on what could be hoped for.

Based on joint work with Henri Lombardi and Daniel Perrucci.

Contact
Name(s):Michael Sagraloff
Video Broadcast
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Created by:Michael Sagraloff, 02/05/2016 12:07 PMLast modified by:Uwe Brahm/MPII/DE, 11/24/2016 04:13 PM
  • Michael Sagraloff, 02/12/2016 10:37 AM
  • Michael Sagraloff, 02/05/2016 12:07 PM -- Created document.