On intrinsic bounds in the Nullstellensatz
Suppose \(A=k [X_1, \dots, X_n]\) be a polynomial ring over a field \(k\) and let \(f_1,f_2, \dots, f_s\) be a trivial system in \(A\). (That means they generate the unit ideal.) The geometric degree \(\delta\) associated to such a trivial system was defined in this paper, which we give later. The main result in this paper is as follows: Suppose \(f_1,f_2, \dots, f_s\) is a trivial system in \(A\). Let \(\delta\) be the associated geometric degree and let \(d= \max_j \deg f_j\). Let \(D(f_1, \dots, f_s)= \min \{\max \{\deg p_j f_j;1 =\sum p_i f_i\}\}\). Then \[ D(f_1, \dots, f_s) \leq 2n^2\delta d. \] If the characteristic of \(k\) is zero, then the geometric degree is defined as follows. Let \({\mathcal G}\) be set of all sequences \(g_1, \dots, g_t\) where \(2\leq t\leq n+1\) such that (1) \(g_1, \dots, g_t\) is a trivial system and each \(g_i\) is a \(\overline k\)-linear combination of \(f_1, \dots, f_s\), (2) \(g_1, \dots, g_{t-1}\) is a regular sequence, (3) \((g_1, \dots, g_j)\) is a radical ideal, for \(j=1, \dots,t\). The set \({\mathcal G}\) is nonempty by Bertini's theorem. The geometric degree \(\delta\) is defined as: \[ \delta =\min_{\mathcal G} \{\max_{1\leq j\leq \min t-1, n-1} \{\deg V(g_1, \dots, g_j)\}\}. \]
scientific article; zbMATH DE number 998035