On the number of polynomials and integral elements of given discriminant

Abstract

Let K be a field of characteristic 0, let R be a subring of K which has K as its quotient field, let G be a finite, normal extension of Kand let R' be an integral extension ring of R in G. We shall suppose that either R is finitely generated over Z (we shall refer to this as the absolute case) or R is finitely generated over a field k of characteristic 0 which is algebraically closed in K (this will be called the relative ca!e). Let n~2 be an integer. By tfi(n, R, R') we shall denote the set of all polynomials f(X)ER [X] of degree n which are monic and all of whose zeros are simple and belong to R'. By tfi(R, R') we denote the set U .P(n, R, R'). Let f3 be a fixed, n~2 non-zero element of R. We shall study the sets of polynomials f (X)E t/J(R, R')l~i<j; §n We call two polynomials f(X), g(X)ER[X] R-equivalent if g(X)=f(X+a) for some aER and weakly R-equivalent if g(X)=ud•sf f(X/u+a) for some uER* and aE R. The corresponding equivalence classes will be called R-equivalence classes and weak R-equivalence classes, respectively. If two polynomialsf, g are R-equivalent then D(f)=D(g) whereas iff, g are weakly R-equivalent then D(f)=eD(g) with some eER*.In the absolute case Gyory [6], [1] proved that if R is integrally closed in K then the polynomials .f(X)E <P(R, R') which satisfy (1) belong to at most finitely many R-equivalence classes and the polynomialsf(X)EiP(R, R') satisfying (2) belong to at most finitely many weak R-equivalence classes. Further, in [8] he showed that these equivalence classes can be determined effectively provided that R, K, G, R' and fJ are given explicitly in a certain well-defined sense (cf. [8], § 2.1). As consequences, in [8] ( cf. also [9]) he obtained effective finiteness theorems for integral elements with *The research was done at the University of Leiden in the academic year 1983/1984. 1 " If Risa ring, then R* denotes its group of units and R+ its additive group.

Bounds for the solutions of decomposable form equations

“…Under the assumptions of Theorem 3, m = deg(f 0 ) can be estimated from above (cf. [8]), Theorem 2) by an explicit bound which depends only on d, s and the number of distinct prime ideal divisors of β.…”

S-unit equations and their applications

“…Corollary 17.1 (Evertse and Gyory [23] For explicit expressions for c1 to c4 and for references, see Gyory [31], [33). The results presented above have various generalisations; for references see [45], [46], [23].…”

“…The results presented above have various generalisations; for references see [45], [46], [23]. is not a root of unity for 1 ~ i < j :::; n, then g( a1 ), ... , g( an) are algebraically independent.…”

Polynomials and binary forms with given discriminant

“…Theorem C from [15] and the results of [14], [12] and [18] have several applications. We mention some of them in their simplest form; more general and quantitative versions can be found in [14] and [15].…”

“…Evertse and the author [15] derived explicit upper bounds for the number of equivalence classes of monic polynomials with integral coefficients and given non-zero discriminant. We denote by ω(D) the number of distinct prime factors of a non-zero integer D.…”