ISSN:
1432-0622
Keywords:
Complete intersection polynomial ideals
;
Trace theory
;
Bezout's inequality
;
Effective Nullstellensatz
;
Bertini's theorem
Source:
Springer Online Journal Archives 1860-2000
Topics:
Computer Science
,
Mathematics
,
Technology
Notes:
Abstract In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Letk be an infinite perfect field and let f1,...,f n−r∈k[X1,...,Xn] be a regular sequence with d:=maxj deg fj. Denote byA the polynomial ringk [X1,..., Xr] and byB the factor ring k[X1,...,Xn]/(f1,...,fn r); assume that the canonical morphism A→B is injective and integral and that the Jacobian determinantΔ with respect to the variables Xr+1,...,Xn is not a zero divisor inB. Let finally σ∈B*:=HomA(B, A) be the generator of B* associated to the regular sequence. We show that for each polynomialf the inequality deg σ(¯f) ≦dn r(δ+1) holds (¯fdenotes the class off inB andδ is an upper bound for (n−r)d and degf). For the usual trace associated to the (free) extensionA ↪B we obtain a somewhat more precise bound: deg Tr(¯f) ≦ dn r degf. From these bounds and Bertini's theorem we deduce an elementary proof of the following effective Nullstellensatz: let f1,..., fs be polynomials in k[X1,...,Xn] with degrees bounded by a constant d≧2; then 1 ∈(f1,..., fs) if and only if there exist polynomials p1,..., ps∈k[X1,..., Xn] with degrees bounded by 4n(d+ 1)n such that 1=Σipifi. in the particular cases when the characteristic of the base fieldk is zero ord=2 the sharper bound 4ndn is obtained.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01198015
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