An analytic approach to the degree bound in the Nullstellensatz (Q2790190)

Given \(n\) polynomials without a common zero in the complex plane \(\mathbb{C}\), \(p_j\in\mathbb{C}[z]\), \(1\leq j\leq n\), one can find polynomials \(q_j\in\mathbb{C}[z]\), \(1\leq j\leq n\), such that NEWLINE\[NEWLINE\sum_{j=1}^n p_jq_j=1,NEWLINE\]NEWLINE because of Hilbert's Nullstellensatz in the one-variable case.NEWLINENEWLINEIt has being of interest to obtain the best bound for the degrees of such polynomials \(q_j\). Until \textit{W. D. Brownawell} [Ann. Math. (2) 126, 577--591 (1987; Zbl 0641.14001)], it was widely expected that this issue exhibited double exponential growth. In the paper under review, via a simple analytic method, the authors state the result by Brownawell in a special case:NEWLINENEWLINETheorem. Let \(p_j\in\mathbb{C}[z]\), \(1\leq j\leq n\), be \(n\) polynomials without a common zero in \(\mathbb{C}\) and let \(\mathrm{deg}(p_j)\leq D\). Then there exist \(n\) polynomials \(q_j\in\mathbb{C}[z]\), \(1\leq j\leq n\), satisfying NEWLINE\[NEWLINE\sum_{j=1}^n p_jq_j=1NEWLINE\]NEWLINE and such that \(\mathrm{deg}(q_j)\leq D-1\).