Finite free resolutions of length two and koszul generalized complex |
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Authors: | J A Hermida-Alonso T Sánchez-Giralda |
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Institution: | Departamento de Algebra,Facultad de Ciencias , Universidad de Valladolid , Valladolid, 47005, Spain |
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Abstract: | Abstract In 1956, Ehrenfeucht proved that a polynomial f 1(x 1) + · + f n (x n ) with complex coefficients in the variables x 1, …, x n is irreducible over the field of complex numbers provided the degrees of the polynomials f 1(x 1), …, f n (x n ) have greatest common divisor one. In 1964, Tverberg extended this result by showing that when n ≥ 3, then f 1(x 1) + · + f n (x n ) belonging to Kx 1, …, x n ] is irreducible over any field K of characteristic zero provided the degree of each f i is positive. Clearly a polynomial F = f 1(x 1) + · + f n (x n ) is reducible over a field K of characteristic p ≠ 0 if F can be written as F = (g 1(x 1)) p + (g 2(x 2)) p + · + (g n (x n )) p + cg 1(x 1) + g 2(x 2) + · + g n (x n )] where c is in K and each g i (x i ) is in Kx i ]. In 1966, Tverberg proved that the converse of the above simple fact holds in the particular case when n = 3 and K is an algebraically closed field of characteristic p > 0. In this article, we prove an extension of Tverberg's result by showing that this converse holds for any n ≥ 3. |
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Keywords: | Polynomials (factorization) Polynomials (irreducibility) Special polynomials |
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