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