On the effective Nullstellensatz (Q2571047)

Given finitely many polynomials \(f_1,\dots,f_k\) in the coordinate ring \(\mathbb K[X]\) of an \(n\)-dimensional affine variety \(X\subset\mathbb K^m\) over an algebraically closed field \(K\) such that \(f_1,\dots,f_k\) have no common zero in \(X\), the effective version of Hilbert's Nullstellensatz asks for an equation \(1 = \sum^k_{i=1}f_ig_i\) with an explicit bound for \(\deg(f_ig_i)\) in terms of the degree \(D\) of \(X\), the dimension \(n = \dim(X)\), and the degrees \(d_i = \deg(f_i)\). For \(X =\mathbb K^n\), sharp estimates in almost all cases were proven by \textit{J. Kollár} [J. Am. Math. Soc. 1, 963--975 (1988; Zbl 0682.14001)] and \textit{M. Sombra} [Adv. Appl. Math. 22, No.~2, 271--295 (1999; Zbl 0933.14001)]. The author partially improves and generalizes these bounds by showing that, for any \(X\), it is possible to have \(\deg(f_ig_i)\leq DN'(d_1,\dots,d_k; n)\) for \(k\leq n\) and \(\deg f_ig_i)\leq 2DN'(d_1,\dots,d_k; n) - 1\) for \(k > n\). Here we assume \(d_1\geq\cdots\geq d_k\) and \(N'(d_1,\dots,d_k;n)\) equals \(d_i\) for \(n = 1\), it is \(d_1\cdots d_k\) for \(n > k > 1\), and it is \(d_1\cdots d_{n-1}\cdots d_k\) for \(k > n > 1\). The proof uses a generalization of the classical Perron theorem, the author's characterization of the set of non-properness of a generically finite map \(f : X\to \mathbb K^n\) [see \textit{Z. Jelonek}, Bull. Pol. Acad. Sci. J. Math. 49, No.~3, 279--283 (2001; Zbl 1065.14074)], and a generalized version of the elimination theorem. Next, the author proves a bound on the Noether exponent \(e(I)\) of the ideal \(I =\langle f_1,\dots,f_k\rangle\) in \(\mathbb [X]\). He shows \(e(I)\leq DN(d_1,\dots, d_k;n)\) where \(N(d_1,\dots,d_k;n)\) equals \(N'(d_1,\dots,d_k;n)\) except for \(N(d_1,\dots,d_k,1) = d_k\). This bound for \(e(I)\) partially improves earlier bounds of \textit{J. Kollár} in [J. Eur. Math. Soc. (JEMS) 1, No.~3, 313--337 (1999; Zbl 0986.14043)], by \textit{M. Sombra} [loc. cit.], and by \textit{L. Ein} and \textit{R. Lazarsfeld} [Invent. Math. 137, No.~2, 427--448 (1999; Zbl 0944.14003)]. Finally, the author applies bis bounds to derive a bound for the representation \(\Phi(x_i) = \sum^n_{j=1} g_{ij}f_j\) of the \(i\)-th elimination polynomial if \(X = \mathbb K^n\), \(k = n\), and the set of zeros of \(\langle f_1,\dots,f_n\rangle\) is finite. Using suitable examples, the author verifies that his bounds are sharp in many cases.