B Ezout Identities with Inequality Constraints

We dedicate this paper to Ambikeshwar Sharma with esteem, admiration and gratitude for lighting our way. Abstract. This paper examines the set B(P) = fQ : P Q = 1 ; Q 2 R m g where P 2 R m is unimodular and R is either the algebra P R of algebraic polynomials which are real{valued on the cube I d or the algebra L R of Laurent polynomials which are real{valued on the torus T d : We sharpen previous results for the case m = 2; d = 1 by showing that if P is non-negative, then there exists a positive Q 2 B(P) whose length is bounded by a function of the length of P and the separation between the zeros of P: In the general case we employ the Quillen-Suslin theorem, the Swan theorem, the Weierstrass approximation theorem and the Michael selection theorem to prove a result about the existence of solutions to the B ezout identity with inequality constraints.