The Prouhet–Tarry–Escott problem, indecomposability of polynomials and Diophantine equations

Abstract

In this paper, we show how the subjects mentioned in the title are related. First we study the structure of partitions of $$A \subseteq \{1, \dots , n\}$$ A ⊆ { 1 , ⋯ , n } in k-sets such that the first $$k-1$$ k - 1 symmetric polynomials of the elements of the k-sets coincide. Then we apply this result to derive a decomposability result for the polynomial $$f_A(x) := \prod _{x \in A} (x-a)$$ f A ( x ) : = ∏ x ∈ A ( x - a ) . Finally we prove two theorems on the structure of the solutions (x, y) of the Diophantine equation $$f_A(x)=P(y)$$ f A ( x ) = P ( y ) where $$P(y)\in \mathbb {Q}[y]$$ P ( y ) ∈ Q [ y ] and on shifted power values of $$f_A(x)$$ f A ( x ) .

The Diophantine equation f(x)=g(y)$f(x)=g(y)$ for polynomials with simple rational roots

“…For papers with $$K-k \geqslant 2, b_{\ell }=0$$ see [3] and [22]. Hajdu, Papp and Tijdeman [41] provided effective upper bounds for $$\max (|x|,|y|,\ell )$$ in () under the assumption that $$K-k < cK^{2/3}$$ for some explicit $$c>0$$.…”

“…Kulkarni and Sury [48] for $$d=1$$, $$k=K$$, $$\ell >2$$ completely described all cases where () has infinitely many solutions. Hajdu, Papp and Tijdeman [41] proved the finiteness of the number of solutions of () under the assumption that $$K-k \leqslant cK^{2/3}$$ with $$c$$ an explicit constant, except for two explicitly given classes of $$g$$s. Both papers are based on a theorem of Bilu and Tichy [11], which will also play an important role in our present study and is formulated in the next section.…”

The Diophantine equation $f(x)=g(y)$ for polynomials with simple rational roots

“…Mukhopadhyay and Shorey [53] for ℓ = 2 and Saradha and Shorey [66] for ℓ ≥ 3 determined all solutions of equation ( 5) with [3] and [20]. Hajdu, Papp and Tijdeman [39] provided effective upper bounds for max(|x|, |y|, ℓ) in ( 5) under the assumption that K − k < cK 2/3 for some explicit c > 0.…”

“…. , a N be distinct rationals, and assume that with some rational numbers Note that N ≤ 12 is already proved in [39] (see the proof of Theorem 2.3 there). However, in this paper we need a more precise statement.…”

“…Hajdu, Papp and Tijdeman [39] proved the finiteness of the number of solutions of (9) under the assumption that K − k ≤ cK 2/3 with c an explicit constant, provided that g does not belong to two explicitly given classes in which there can be infinitely many solutions. The latter two papers are based on a theorem of Bilu and Tichy [11], which will also play an important role in our present study and is formulated in the next section.…”