Bounds for traces in complete intersections and degrees in the Nullstellensatz
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,......
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1995
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| Acceso en línea: | https://bibliotecadigital.exactas.uba.ar/collection/paper/document/paper_09381279_v6_n6_p353_Sabia http://hdl.handle.net/20.500.12110/paper_09381279_v6_n6_p353_Sabia |
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| Sumario: | 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. |
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