Tuesday, August 30 |
07:00 - 08:45 |
Breakfast ↓ Breakfast is served daily between 7 and 9am in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
09:00 - 09:25 |
Jan-Hendrik Evertse: Effective results for Diophantine equations over finitely generated domains (I) ↓ We consider Diophantine equations with unknowns taken from finitely generated domains of characteristic 0. Up to isomorphism, such a domain is of the shape Z[X1,…,Xr]/I, where I is a prime ideal of Z[X1,…,Xr] with I∩Z=(0). Special cases of such domains are rings of (S−)-integers in number fields and polynomial rings over Z.
Lang (1960) was the first to prove finiteness results over arbitrary finitely generated domains of characteristic 0 for various classes of Diophantine equations, by combining Roth's theorem over number fields, Roth's theorem over function fields, and specialization arguments. His finiteness results were ineffective, in that their proofs did not provide methods to determine all solutions. Győry (1983/84) proved effective finiteness results for certain classes of Diophantine equations, valid for a restricted class of finitely generated domains. Later, Győry and E. (2013) managed to generalize Győry's effective method to arbitrary finitely generated domains of characteristic 0.
The idea is to map the equation over the finitely generated domain under consideration
to various related equations over rings of S-integers in number fields by means of specializations and use effective height estimates for the solutions of the equations over the S-integers, e.g., obtained by means of Baker's method. Which specializations to use is controlled by height estimates for the solutions of related equations over function fields. In 2013, Győry and E. obtained in this manner an effective finiteness result for unit equations ax+by=c in x,y∈A∗, with A any finitely generated domain of characteristic 0. This was extended later to various other classes of Diophantine equations over finitely generated domains.
In my talk I would like to give an idea how the method of Győry and E. works and give some applications. This is a prequel to the talk of Attila Bérczes.
Reference:
J.-H. Evertse, K. Győry: Effective results and methods for Diophantine equations over finitely generated domains, London Math. Soc. Lecture Note Ser. 475. (Online) |
09:30 - 09:55 |
Attila Bérczes: Effective results for Diophantine equations over finitely generated domains ↓ Let A:=Z[z1,…,zr]⊃Z be a finitely generated integral domain over
Z and denote by K the quotient field of A. Finiteness results for several kinds of Diophantine equations over A date back to the middle of the last century. S. Lang generalized several earlier results on Diophantine equations over the integers to results over A, including results concerning unit equations, Thue-equations and integral points on curves. However, all his results were ineffective.
The first effective results for Diophantine equations over finitely generated domains were published in the 1980's, when Győry developed his new effective specialization method. This enabled him to prove effective results over finitely generated domains of a special type.
In 2011 Evertse and Győry refined the method of Győry such that they were able to prove effective results for unit equations ax+by=1 in x,y∈A∗ over arbitrary finitely generated domains A of characteristic 0. Using this new general method Bérczes, Evertse and Győry obtained effective results for Thue equations, hyper- and superelliptic equations and for the Schinzel-Tijdeman equation over arbitrary finitely generated domains. Koymans generalized the effective result of Tijdeman on the Catalan equation for finitely generated domains, while Evertse and Győry proved effective results for decomposable form equations in this generality. Bérczes proved effective results for equations F(x,y)=0 in x,y∈A∗ for arbitrary finitely generated domains A, and for F(x,y)=0 in x,y∈¯Γ, where F(X,Y) is a bivariate polynomial over A and ¯Γ
is the division group of a finitely generated subgroup Γ of K∗.
In my talk I will focus mainly on these latter mentioned results, a short survey
how the method of Evertse and Győry could be used in the proof of these results. (TCPL 201) |
10:00 - 10:30 |
Coffee Break (TCPL Foyer) |
10:30 - 10:55 |
Kálmán Györy: Bounds for the solutions of S-unit equations in two unknowns over number fields ↓ The S-unit equations in two unknowns, equations of the form
\begin{align*}
\alpha x+\beta y=1,
\end{align*}
where the unknowns x,y are S-units in a number field K containing \alpha,\beta, are very important in the solution of many other families of Diophantine equations. For their application to obtaining the complete solution of Diophantine equations, an upper bound on the (height of) solutions of associated S-unit equations is required.
The speaker (1974,79) gave explicit upper bounds for (slightly more general) solutions of S-unit equations, and used them to get various applications. Later several authors, including Evertse, Stewart, Tijdeman and Gy (1988), Bombieri (1993), Bugeaud and Gy (1996), Bugeaud (1998), Yu and Gy (2006) and Evertse and Gy (2015) improved upon or modified the previous bounds. Their bounds depend among others on the cardinality |S| of S and the largest norm P of the prime ideals in S.
In Yu and Gy (2006) we obtained two different, considerably improved bounds for the solutions. Le Fourn (2020) combined the proof of the first bound with his variant of Runge's method to replace P in the first bound by the third largest norm P' of the prime ideals in S. In Gy (2020) we refined the second, more complicated proof and combined it with Le Fourn's idea to replace P in the second bound by P'. Further, we improved also the dependence on |S| and, in terms of S, derived the best known bound to date for the solutions.
In our talk we formulate the bounds from Gy (1979), Bugeaud and Gy (1996), Yu and Gy (2006), Le Fourn (2020), and Gy (2020), compare the bounds, emphasize the main tool and outline the main steps in the proof of Gy (2020). Further, we present some recent applications of our latest bound, giving improved upper bound on the S-integral solutions of Thue equations and some more general decomposable form equations over number fields, and providing the best Masser's type ABC inequality to date towards Masser's ABC conjecture over number fields; Gy (2022). (Online) |
11:00 - 11:25 |
Robert Tijdeman: Diophantine equations f(x)=g(y) with infinitely many rational solutions ↓ A theorem of Bilu and Tichy (2000) gives deep insight into the structure of polynomials f,g with rational coefficients such that the equation f(x) = g(y) has infinitely many rational solutions x,y. Lajos Hajdu and I have worked out the consequences in case f has only simple rational roots, a case often considered in the literature. We describe the pairs (deg(f), deg(g)) which are possible, also in case both f and g have only simple rational roots. There is a connection with the classical Prouhet-Tarry-Escott problem to find two disjoint sets of n integers such that the sums of the k-th powers are equal for k<n. (Online) |
11:30 - 13:00 |
Lunch ↓ Lunch is served daily between 11:30am and 1:30pm in the Vistas Dining Room, the top floor of the Sally Borden Building. (Vistas Dining Room) |
13:00 - 14:30 |
Alina Ostafe: Discussions (Online) |
14:30 - 14:55 |
Peter Koymans: The negative Pell equation and applications ↓ In this talk we will study the negative Pell equation, which is the conic C_D : x^2 - D y^2 = -1 to be solved in integers x, y \in \mathbb{Z}. We shall be concerned with the following question: as we vary over squarefree integers D, how often is C_D soluble? Stevenhagen conjectured an asymptotic formula for such D. Fouvry and Kluners gave upper and lower bounds of the correct order of magnitude. We will discuss a proof of Stevenhagen's conjecture, and potential applications of the new proof techniques. This is joint work with Carlo Pagano. (TCPL 201) |
15:00 - 15:25 |
Fabrizio Barroero: On the polynomial Pell equation ↓ We call a complex polynomial D “pellian” if there are non-constant polynomials A and B such that A^2-DB^2=1. While all non-square quadratic polynomials are pellian, there are square-free polynomials of any even degree \geq 4 that are not pellian. Masser and Zannier considered one-parameter families of polynomials which are non-identically pellian and studied the pellian specialisations. They gave a criterion for the existence of infinitely many pellian specialisation.
In joint work with Laura Capuano and Umberto Zannier we consider the “moduli space” of monic polynomials of fixed even degree 2d \geq 4 and prove, among other things, that the locus of pellian polynomials consists of a denumerable union of subvarieties of dimension at most d+1. (TCPL 201) |
15:30 - 16:00 |
Coffee Break (TCPL Foyer) |
16:00 - 16:25 |
Laura Capuano: Multiplicative and linear dependence over finite fields and on elliptic curves modulo primes ↓ Given n multiplicatively independent rational functions f_1, \ldots, f_n with rational coefficients, there are at most finitely many complex numbers a such that f_1(a), \ldots, f_n(a) satisfy two independent multiplicative relations. This was proved independently by Maurin and by Bombieri, Habegger, Masser and Zannier, and it is an instance of more general conjectures of unlikely intersections over tori made by Bombieri, Masser and Zannier and independently by Zilber. We consider a positive characteristic variant of this problem, proving that, for sufficiently large primes, the cardinality of the set of a \in \mathbb F_p such that f_1(a), \ldots, f_n(a) satisfy two independent multiplicative relations with exponents bounded by a constant K is bounded independently of K and p. We prove also analogous results for products of elliptic curves and for split semiabelian
varieties. This is a joint work with F. Barroero, L. Mérai, A. Ostafe and M. Sha. (TCPL 201) |
16:30 - 16:55 |
László Mérai: Divisors of sums of polynomials ↓ In a series of papers, Sárközy and Stewart studied the prime divisors of sum-sets \mathcal{A}+\mathcal{B}. Among others, they showed that if \mathcal{A},\mathcal{B}\subset \{1,\dots, N\} are not too small, then there are a\in\mathcal{A} and b\in\mathcal{B} such that a+b has large prime divisors.
In this talk we explore this problem for polynomials over finite fields. In particular, we show that if \mathcal{A},\mathcal{B} \subset \mathbb{F}_q[x] are sets of polynomials of degree n, then a+b has large degree irreducible divisors for some a\in\mathcal{A}, b\in\mathcal{B}. In particular, if \mathcal{A},\mathcal{B} have positive relative densities, then a+b has an irreducible divisor of degree n+O(1) for some a\in\mathcal{A}, b\in\mathcal{B}. (TCPL 201) |
17:30 - 19:30 |
Dinner ↓ A buffet dinner is served daily between 5:30pm and 7:30pm in Vistas Dining Room, top floor of the Sally Borden Building. (Vistas Dining Room) |