Real Algebraic and Analytic Geometry |

Lower bounds for a polynomial in terms of its coefficients.

e-mail: ,

Submission: 2009, December 30.

*Abstract:
Recently Lasserre gave sufficient conditions in terms of the coefficients for a
polynomial f of degree 2d (d >= 1) in n variables to be a sum of squares of
polynomials. Exploiting this result, we are able to determine, for any
polynomial f of degree 2d whose highest degree term is an interior point in the
cone of sums of squares of forms of degree d, a real number r such that f-r is a
sum of squares of polynomials. The existence of such a number r was proved
earlier by Marshall, but no estimates for r were given. We also determine a
lower bound for any polynomial f whose highest degree term is positive
definite. Similar arguments are applied to results of Fidalgo and Kovacec, to
determine other numbers r such that f-r is a sum of squares and other lower
bounds for f.*

Mathematics Subject Classification (2000): 12D99 14P99 90C22.

Keywords and Phrases: Positive polynomials, sums of squares, optimization.

**Full text**, 8p.:
dvi 45k,
ps.gz 273k,
pdf 305k.

Server Home Page