Consider the general Bézout equation of the form \(A_ 1X_ 1+...+A_ rX_ r=C\) where C and the \(A_ i\) are from a polynomial ring R, and we are looking for a solution for the unknowns \(X_ i\) in the same ring. The case where R is the ring of polynomials in two variables over the real or complex field and \(C=1\) arises in multidimensional systems and control theory [see for example, \textit{T. Kailath}, ``Linear systems'' (1980; Zbl 0454.93001)]. The main result of the present paper (theorem 8) generalizes a result of \textit{M. Šebek} [Kibernetika 19, 212-224 (1983; Zbl 0515.93036)] on bounding the degrees of the polynomials in the solution. Let \(R=D[w]\) be a polynomial ring in one indeterminate w over an integral domain D, and suppose that the Bezout equation has at least one solution. Suppose that the degrees in w of \(A_ i\) and C are \(d_ i\) and d, respectively, and that the ideal generated by the leading coefficients of \(A_ i\) and \(A_ j\) is D for all \(i\neq j\). Let \(\bar d\) denote the maximum of all \(d_ i+d_ j\quad (i\neq j)\quad and\quad d.\) Then there is a solution of the equation with \(\deg (X_ i)\leq \bar d- d_ i\) for each i. \((Theorem\quad 8\) is actually stated in a more restricted form in the paper but, as the authors point out, the proof given does not require the extra hypotheses.) Some analogous results are given when R is the ring of entire analytic functions on \({\mathbb{C}}\) and the \(A_ i\) and C have a special form.