On the degrees of bases of free modules over a polynomial ring

Let k be an infinite field, A the polynomial ring k[x1 , . . . , xn] and F ∈ AN×M a matrix such that Im F ⊂ AN A-free (in particular, Quillen-Suslin Theorem implies that Ker F is also free). Let D be the maximum of the degrees of the entries of F and s the rank of F. We show that there exists a basi...

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Detalles Bibliográficos
Autores principales: Almeida, M., D'Alfonso, L., Solerno, P.
Formato: JOUR
Acceso en línea:http://hdl.handle.net/20.500.12110/paper_00255874_v231_n4_p679_Almeida
Aporte de:
Biblioteca Digital - Facultad de Ciencias Exactas y Naturales (UBA) de Universidad de Buenos Aires
Descripción
Sumario:Let k be an infinite field, A the polynomial ring k[x1 , . . . , xn] and F ∈ AN×M a matrix such that Im F ⊂ AN A-free (in particular, Quillen-Suslin Theorem implies that Ker F is also free). Let D be the maximum of the degrees of the entries of F and s the rank of F. We show that there exists a basis {v1 , . . . , vM} of AM such that {v1 , . . . , vM-s} is a basis of Ker F, {F(vM-s+1), . . . , F(vM)} is a basis of Im F and the degrees of their coordinates are of order ((M - s)sD)O(n4). This result allows to obtain a single exponential degree upper bound for a basis of the coordinate ring of a reduced complete intersection variety in Noether position.

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