Polynomials and binary forms with given discriminant

Abstract

There is an extensive literature of monic polynomials and binary forms with given discriminant. The first part of our paper gives a brief survey of the most important results over Z on such polynomials and binary forms as well as on their various applications. In the second part we improve some earlier general effective and quantitative results over number fields. As an application we obtain some new information about the arithmetical properties of discriminants of polynomials and binary forms.

Distances between the conjugates of an algebraic number

“…By an elementary argument one shows that ( * ) {i,j}⊂Σ |α (i) − α (j) | C • M (α) −κ holds for all α with K = Q(α), with C = 2 −r(r−1)/2 and κ = r − 1. In the present paper we deduce inequalities ( * ) with κ < r − 1 and with a constant C depending on K which are valid for all α with Q(α) = K. We obtain such inequalities with an ineffective constant C, using arguments and results from [7], [8], and with an effective constant C using a result from [9], [11].Define κ(Σ) to be the infimum of all real numbers κ for which there exists a constant C > 0 such that ( * ) holds for every α with Q(α) = K. Then clearly κ(Σ) r − 1. We describe the sets Σ for which κ(Σ) = r − 1 and we give upper bounds for κ(Σ) in case that it is smaller than r − 1.…”

“…By an elementary argument one shows that ( * ) {i,j}⊂Σ |α (i) − α (j) | C • M (α) −κ holds for all α with K = Q(α), with C = 2 −r(r−1)/2 and κ = r − 1. In the present paper we deduce inequalities ( * ) with κ < r − 1 and with a constant C depending on K which are valid for all α with Q(α) = K. We obtain such inequalities with an ineffective constant C, using arguments and results from [7], [8], and with an effective constant C using a result from [9], [11].…”

“…Proof. Lemma 3.2 is precisely Corollary 5 of [11] (this is in fact a slight improvement of a special case of Theorem 3' of [9]). As for Lemma 3.1, to each α with Q(α) = K we associate the binary form…”

“…To prove Theorem 1.3, we use a recent result by Győry [11] (which is a slight improvement of an older result by Győry and the author [9]) stating in a precise form that every algebraic number α is equivalent to a number α * with…”

Power values of polynomials and binomial Thue--Mahler equations

“…In the proof of Theorem 3 we utilize among other things some new estimates of Matveev [16] and Yu [25] on linear forms in logarithms of algebraic numbers and a recent bound of Győry and Yu [13] on the solutions of S-unit equations. Theorem 2 will be deduced from Theorem 3 with the help of a recent effective theorem of Győry [12] concerning monic binary forms having discriminants contained in S. We remark that Theorems 1 and 2 can be proven, with other bounds, without the use of Theorem 3 as well. This will be the subject of a forthcoming paper.…”