The algebraic structure of the set of solutions to the Thue equation

AbstractLet Fn be a binary form with integral coefficients of degree n⩾2, let d denote the greatest common divisor of all non-zero coefficients of Fn, and let h⩾2 be an integer. We prove that if d=1 then the Thue equation (T) Fn(x,y)=h has relatively few solutions: if A is a subset of the set T(Fn,h) of all solutions to (T), with r:=card(A)⩾n+1, then(#)h divides the number Δ(A):=∏1⩽k<l⩽rδ(ξk,ξl), where ξk=〈xk,yk〉∈A, 1⩽k⩽r, and δ(ξk,ξl)=xkyl−xlyk. As a corollary we obtain that if h is a prime number then, under weak assumptions on Fn, there is a partition of T(Fn,h) into at most n subsets maximal with respect to condition (#)

Representation of integers by sparse binary forms

We will give new upper bounds for the number of solutions to the inequalities of the shape $|F(x , y)| \leq h$, where $F(x , y)$ is a sparse binary form, with integer coefficients, and $h$ is a sufficiently small integer in terms of the absolute value of the discriminant of the binary form $F$. Our bounds depend on the number of non-vanishing coefficients of $F(x , y)$. When $F$ is really sparse, we establish a sharp upper bound for the number of solutions that is linear in terms of the number of non-vanishing coefficients. This work will provide affirmative answers to a number of conjectures posed by Mueller and Schmidt in 1988, for special but important cases

Polynomial Root Distribution and Its Impact on Solutions to Thue Equations

In this study, we focus on two topics in classical number theory. First, we examine Thue equations—equations of the form F(x, y) = h where F(x, y) is an irreducible, integral binary form and h is an integer—and we give improvements to both asymptotic and explicit bounds on the number of integer pair solutions to Thue equations. These improved bounds largely stem from improvements to a counting technique associated with “The Gap Principle,” which describes the gap between denominators of good rational approximations to an algebraic number. Next, we will take inspiration from the impact of polynomial root distribution on solutions to Thue equations and we examine polynomial root distribution as its own topic. Here, we will look at the relation between the separation of a polynomial—the minimal distance between distinct roots—and the Mahler measure of a polynomial—a height function which connects the roots of a polynomial with its coefficients. We make a conjecture about how separation can be bounded above by the Mahler measure and we give data supporting that conjecture along with proofs of the conjecture in some low-degree cases

30 years of collaboration

We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)