Bounds for traces in complete intersections and degrees in the Nullstellensatz |
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Authors: | Juan Sabia Pablo Solernó |
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Affiliation: | (1) Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina |
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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–rk[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,...,fnr); assume that the canonical morphism AB is injective and integral and that the Jacobian determinant with respect to the variables Xr+1,...,Xn is not a zero divisor inB. Let finally B*:=HomA(B, A) be the generator of B* associated to the regular sequence.We show that for each polynomialf the inequality deg (¯f) dnr(+1) holds (¯fdenotes the class off inB and 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) dnr 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 d2; then 1 (f1,..., fs) if and only if there exist polynomials p1,..., psk[X1,..., Xn] with degrees bounded by 4n(d+ 1)n such that 1=ipifi. in the particular cases when the characteristic of the base fieldk is zero ord=2 the sharper bound 4ndn is obtained.Partially supported by UBACYT and CONICET (Argentina) |
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Keywords: | Complete intersection polynomial ideals Trace theory Bezout's inequality Effective Nullstellensatz Bertini's theorem |
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