Reducibility of Polynomials in Two Variables
نویسندگان
چکیده
منابع مشابه
Reducibility of Polynomials
Reducibility without qualification means in this lecture reducibility over the rational field. Questions on such reducibility occupy an intermediate place between questions on reducibility of polynomials over an algebraically closed field and those on primality. I shall refer to these two cases as to the algebraic and the arithmetic one and I shall try to exhibit some of the analogies consideri...
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Introduction Let K be a number field, ax the ring of integers of K. Suppose we are given a projective curve 2, and a morphism 4 : Z 4 Pf(C) (where Pf(C) is the projective line) such that 4 and Z are both defined over K. We denote by K(Z) the field of functions of Z defined over K, and from this data we obtain a permutation representation T of the Galois group G(K(Z) "/K(Pt(C))) (where K(Z) A is...
متن کاملWaring’s Problem for Polynomials in Two Variables
We prove that all polynomials in several variables can be decomposed as the sums of kth powers: P (x1, . . . , xn) = Q1(x1, . . . , xn) + · · ·+Qs(x1, . . . , xn), provided that elements of the base field are themselves sums of kth powers. We also give bounds for the number of terms s and the degree of the Qi . We then improve these bounds in the case of two variables polynomials of large degre...
متن کاملWaring Problem for Polynomials in Two Variables
We prove that all polynomials in several variables can be decomposed as the sum of kth powers: P (x1, . . . , xn) = Q1(x1, . . . , xn) + · · ·+Qs(x1, . . . , xn), provided that elements of the base field are themselves sum of kth powers. We also give bounds for the number of terms s and the degree of the Qi . We then improve these bounds in the case of two variables polynomials to get a decompo...
متن کاملReducibility of Rational Functions in Several Variables
We prove a analogous of Stein theorem for rational functions in several variables: we bound the number of reducible fibers by a formula depending on the degree of the fraction.
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 1993
ISSN: 0021-8693
DOI: 10.1006/jabr.1993.1063