Univariate Rational Sums of Squares

Teresa Krick (1, 2) , Bernard Mourrain (3) , Agnes Szanto (4)

1 Dep. Mathematica, Universidad Buenos Aires
2 CONICET - Consejo Nacional de Investigaciones Científicas y Técnicas [Buenos Aires]
3 AROMATH - AlgebRe, geOmetrie, Modelisation et AlgoriTHmes
4 Department of Mathematics, North Carolina State University

Given rational univariate polynomials f and g such that gcd(f, g) and f / gcd(f, g) are relatively prime, we show that g is non-negative on all the real roots of f if and only if g is a sum of squares of rational polynomials modulo f. We complete our study by exhibiting an algorithm that produces a certificate that a polynomial g is non-negative on the real roots of a non-zero polynomial f , when the above assumption is satisfied.

Mots clés: Positive polynomials; Sum of Squares; Semi-Definite Matrix; Convex cone; Real roots; Exact computation; Certificate

Domaines: Géométrie algébrique [math.AG]

Teresa Krick, Bernard Mourrain, Agnes Szanto. Univariate Rational Sums of Squares. 2021. ⟨hal-03442193v1⟩