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Abstract:

In this paper we obtain an effective Nullstellensatz using quantitative considerations of the classical duality theory in complete intersections. Let k be an infinite perfect field and let f1,..., f n-r∈k[X1,...,Xn] be a regular sequence with d:=maxj deg fj. Denote by A the polynomial ring k [X1,..., Xr] and by B the factor ring k[X1,...,Xn]/(f1,...,fnr); assume that the canonical morphism A→B is injective and integral and that the Jacobian determinant Δ with respect to the variables Xr+1,...,Xn is not a zero divisor in B. Let finally σ∈B*:=HomA(B, A) be the generator of B* associated to the regular sequence. We show that for each polynomial f the inequality deg σ(-f) ≦dnr(δ+1) holds (-fdenotes the class of f in B and δ is an upper bound for (n-r)d and deg f). For the usual trace associated to the (free) extension A {right arrow, hooked}B we obtain a somewhat more precise bound: deg Tr(-f) ≦ dnr deg f. 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 field k is zero or d=2 the sharper bound 4ndn is obtained. © 1995 Springer-Verlag.

Registro:

Documento: Artículo
Título:Bounds for traces in complete intersections and degrees in the Nullstellensatz
Autor:Sabia, J.; Solernó, P.
Filiación:Departamento de Matemáticas, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires, Buenos Aires, 1428, Argentina
Palabras clave:Bertini's theorem; Bezout's inequality; Complete intersection polynomial ideals; Effective Nullstellensatz; Trace theory
Año:1995
Volumen:6
Número:6
Página de inicio:353
Página de fin:376
DOI: http://dx.doi.org/10.1007/BF01198015
Título revista:Applicable Algebra in Engineering, Communication and Computing
Título revista abreviado:AAECC
ISSN:09381279
CODEN:AAECE
Registro:https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09381279_v6_n6_p353_Sabia

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Citas:

---------- APA ----------
Sabia, J. & Solernó, P. (1995) . Bounds for traces in complete intersections and degrees in the Nullstellensatz. Applicable Algebra in Engineering, Communication and Computing, 6(6), 353-376.
http://dx.doi.org/10.1007/BF01198015
---------- CHICAGO ----------
Sabia, J., Solernó, P. "Bounds for traces in complete intersections and degrees in the Nullstellensatz" . Applicable Algebra in Engineering, Communication and Computing 6, no. 6 (1995) : 353-376.
http://dx.doi.org/10.1007/BF01198015
---------- MLA ----------
Sabia, J., Solernó, P. "Bounds for traces in complete intersections and degrees in the Nullstellensatz" . Applicable Algebra in Engineering, Communication and Computing, vol. 6, no. 6, 1995, pp. 353-376.
http://dx.doi.org/10.1007/BF01198015
---------- VANCOUVER ----------
Sabia, J., Solernó, P. Bounds for traces in complete intersections and degrees in the Nullstellensatz. AAECC. 1995;6(6):353-376.
http://dx.doi.org/10.1007/BF01198015