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Holder-Brascamp-Lieb inequalities provide upper bounds for a class of multilinear expressions, in terms of L^p norms of the functions involved. They have been extensively studied for functions defined on Euclidean spaces. Bennett-Carbery-Christ-Tao have initiated the study of these inequalities for discrete Abelian groups and, in terms of suitable data, have characterized the set of all tuples of exponents for which such an inequality holds for specified data, as the convex polyhedron defined by a particular finite set of affine inequalities. In this paper we advance the theory of such inequalities for torsion-free discrete Abelian groups in three respects.The optimal constant in any such inequality is shown to equal 1 whenever it is finite.An algorithm that computes the admissible polyhedron of exponents is developed. It is shown that nonetheless, existence of an algorithm that computes the full list of inequalitiesin the Bennett-Carbery-Christ-Tao description of the admissible polyhedron for all data,is equivalent to an affirmative solution of Hilbert's Tenth Problem over the rationals.That problem remains open. 

Consider an algorithm computing in a differential field with several commuting derivations such that the only operations it performs with the elements of the field are arithmetic operations, differentiation, and zero testing. We show that, if the algorithm is guaranteed to terminate on every input, then there is a computable upper bound for the size of the output of the algorithm in terms of the size of the input. We also generalize this to algorithms working with models of good enough theories (including, for example, difference fields). We then apply this to differential algebraic geometry to show that there exists a computable uniform upper bound for the number of components of any variety defined by a system of polynomial PDEs. We then use this bound to show the existence of a computable uniform upper bound for the elimination problem in systems of polynomial PDEs with delays. 

Abstract Let $$K$$ be an algebraically closed field of prime characteristic $$p$$ , let $$X$$ be a semiabelian variety defined over a finite subfield of $$K$$ , let $$\unicode[STIX]{x1D6F7}:X\longrightarrow X$$ be a regular self-map defined over $$K$$ , let $$V\subset X$$ be a subvariety defined over $$K$$ , and let $$\unicode[STIX]{x1D6FC}\in X(K)$$ . The dynamical Mordell–Lang conjecture in characteristic $$p$$ predicts that the set $$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$$ is a union of finitely many arithmetic progressions, along with finitely many $$p$$ -sets, which are sets of the form $$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$$ for some $$m\in \mathbb{N}$$ , some rational numbers $$c_{i}$$ and some non-negative integers $$k_{i}$$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $$X$$ is an algebraic torus, we can prove the conjecture in two cases: either when $$\dim (V)\leqslant 2$$ , or when no iterate of $$\unicode[STIX]{x1D6F7}$$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $$X$$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction. 

We study solutions of difference equations in the rings of sequences and, more generally, solutions of equations with a monoid action in the ring of sequences indexed by the monoid. This framework includes, for example, difference equations on grids (for example, standard difference schemes) and difference equations in functions on words. On the universality side, we prove a version of strong Nullstellensatz for such difference equations under the assumption that the cardinality of the ground field is greater than the cardinality of the monoid and construct an example showing that this assumption cannot be omitted. On the undecidability side, we show that the following problems are undecidable: