Sharp effective Nullstellensatz. (English) Zbl 0682.14001

Let K be any field and n, \(d_ 1,...,d_ k\) be natural numbers. Let \(N(n,\underline d)=\min \{s| \quad for\) any homogeneous polynomials \(\phi_ 1,...,\phi_ k\in K[X_ 0,...,X_ n]\) such that \(\deg (\phi_ i)=d_ i\), we have \((rad(I))^ s\subset I\) where \(I=(\phi_ 1,...,\phi_ k)\}.\)
Main theorem: If we assume \(d_ 1\geq d_ 2\geq...\geq d_ k\) and \(d_ i\neq 2\) (for \(all\quad i),\) then N(n,ḏ)\(=d_ 1...d_ k\) if \(k\leq n\), N(n,ḏ)\(=d_ 1...d_{n-1}\cdot d_ k\) if \(k>n>1\), N(n,ḏ)\(=d_ 1+d_ k-1\) if \(k>n=1.\)
Corollary. Let \(f_ 1,...,f_ k, h\in K[X_ 1,...,X_ n]\) and assume that h vanishes on all common zeros of \(f_ 1,...,f_ k\) in the algebraic closure of K. Let \(d_ i=\deg (f_ i)\neq 2\) (for \(all\quad i).\) Then one can find \(g_ 1,...,g_ k\in K[X_ 1,...,X_ n]\) and a natural number s such that \(\sum g_ if_ i =h^ s\), \(s\leq N(n,\underline d)\), \(\deg (g_ if_ i)\leq (1+\deg (h))\cdot N(n,\underline d)\). In particular, if \(h=1\) (so that \(f_ 1,...,f_ k\) have no common zeros), we can choose \(g_ i\) such that \(\sum g_ if_ i =1\) and \(\deg (g_ if_ i)\leq N(n,\underline d).\)
This estimate is the best possible. The condition \(d_ i\neq 2\) is technical, and the author expects that it is not necessary. The proof uses elementary methods of algebraic geometry including local cohomology, and works in all characteristics.
Reviewer: H.Matsumura


14A05 Relevant commutative algebra
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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