Univariate rational sums of squares

Abstract

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 at allthe real roots of f if and only if g is a sum of squares of rational polynomialsmodulo f. We complete our study by exhibiting an algorithm that produces acerticate that a polynomial g is non-negative at the real roots of a non-zeropolynomial f, when the above assumption is satised.