On the minimum of a positive polynomial over the standard simplex

We present a new positive lower bound for the minimum value taken by a polynomial P with integer coefficients in k variables over the standard simplex of R^k, assuming that P is positive on the simplex. This bound depends only on the number of variables k, the degree d and the bitsize @t of the coefficients of P and improves all the previous bounds for arbitrary polynomials which are positive over the simplex.

[1]  M. Mignotte,et al.  Polynomials: An Algorithmic Approach , 1999 .

[2]  Jesús A. De Loera,et al.  An effective version of Pólya's theorem on positive definite forms , 1996 .

[3]  Fabrice Rouillier,et al.  Solving Zero-Dimensional Systems Through the Rational Univariate Representation , 1999, Applicable Algebra in Engineering, Communication and Computing.

[4]  Charles N. Delzell,et al.  Positive Polynomials: From Hilbert’s 17th Problem to Real Algebra , 2001 .

[5]  D. S. Arnon,et al.  Algorithms in real algebraic geometry , 1988 .

[6]  Laureano González-Vega,et al.  Using Symmetric Functions to Describe the Solution Set of a Zero Dimensional Ideal , 1995, AAECC.

[7]  Markus Schweighofer,et al.  On the complexity of Schmu"dgen's Positivstellensatz , 2004, J. Complex..

[8]  M-F Roy,et al.  Géométrie algébrique réelle , 1987 .

[9]  Bernard Mourrain,et al.  The DMM bound: multivariate (aggregate) separation bounds , 2010, ISSAC.

[10]  Pablo Solernó,et al.  Effective Łojasiewicz inequalities in semialgebraic geometry , 1991, Applicable Algebra in Engineering, Communication and Computing.

[11]  B. Reznick,et al.  A new bound for Pólya's Theorem with applications to polynomials positive on polyhedra , 2001 .

[12]  S. Verblunsky,et al.  On Positive Polynomials , 1945 .

[13]  John Canny,et al.  The complexity of robot motion planning , 1988 .

[14]  Marie-Françoise Roy,et al.  A bound on the minimum of a real positive polynomial over the standard simplex , 2009, ArXiv.

[15]  Richard Leroy.  Certificats de positivité et minimisation polynomiale dans la base de Bernstein multivariée. (Certificates of positivity and polynomial minimization in the multivariate Bernstein basis) , 2008 .

[16]  Juan Sabia,et al.  On Sign Conditions Over Real Multivariate Polynomials , 2008, Discret. Comput. Geom..