Wayne M. Lawton  and Charles A. Micchelli
 

Abstract
 This paper examines the set $\Cal B(P) = \{ Q \, : \, P \cdot  Q = 1,\, Q \in \Cal R^{m}\}$, where $P \in \Cal R^{m}$  is unimodular and $\Cal R$ is either the algebra $\Cal P_{R}$ of algebraic polynomials which are real-valued on the cube $\Bbb I^{d}$ or the algebra $\Cal L_{R}$ of Laurent polynomials which are real-valued on the torus $\Bbb T^d.$ 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 \in \Cal 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.