A small value estimate in dimension two involving translations by rational points

Abstract

Abstract. We show that, if a sequence of non-zero polynomials in Z[X 1 , X 2 ] take small values at translates of a fixed point (ξ, η) by multiples of a fixed rational point within the group C×C * , then ξ and η are both algebraic over Q. The precise statement involves growth conditions on the degree and norm of these polynomials as well as on their absolute values at these translates. It is essentially best possible in some range of the parameters.

Linear recurrence sequences and twisted binary forms

“…has the same form as in (8). Therefore the sequence u(a) a∈Z satisfies a linear recurrence relation of order ≤ d. It follows that the conditions…”

“…take prescribed values. Sharp estimates related with this linear system are provided by Lemma 3.1 of [8]. Before stating and proving the next proposition, we introduce the following notation.…”

Multigraded Koszul complexes, filter-regular sequences and lower bounds for the multiplicity of the resultant

“…This construction might be seen as a way to package the information of several auxiliary polynomials f into a single "larger" auxiliary polynomial, namely their resultant. For examples of how the information on the multiplicity of the resultants is used to derive Diophantine results, in the context of interpolation on the commutative algebraic group G a × G m , we refer to [Roy13,Ghi15,NR16].…”