An algorithm is developed for the factorization of a multivariate polynomial represented by traight-line program into its irreducible factors. The algorithm is in random polynomial-time as a function in the input size, total degree, and binary coefficient length for the usual coefficient fields and outputs a straight-line program, which with controllably high probability correctly determines the irreducible factors. It also returns the probably correct multiplicities of each distinct factor. If th oefficient field has finite characteristic p and p divides the multiplicities of some irreducible factors our algorithm constructs straight-line programs for the appropriate p-th powers of such factors. Also a probabilistic algorithm is presented that allows to convert a polynomial given by a straight-line program into its sparse representation. This conversion algorithm is in random-polynomial time in the previously cited parameters and in an upper bound for the number of non-zero...

Diophantine Analysis is a very active domain of mathematical research where one finds more conjectures than results. We collect here a number of open questions concerning Diophantine equations (including Pillai’s Conjectures), Diophantine approximation (featuring the abc Conjecture) and transcendental number theory (with, for instance, Schanuel’s Conjecture). Some questions related to Mahler’s measure and Weil absolute logarithmic height are then considered (e. g., Lehmer’s Problem). We also discuss Mazur’s question regarding the density of rational points on a variety, especially in the particular case of algebraic groups, in connexion with transcendence problems in several variables. We say only a few words on metric problems, equidistribution questions, Diophantine approximation on manifolds and Diophantine analysis on function fields.

Abstract. We study inequalities connecting the product of uniform norms of polynomials with the norm of their product. This circle of problems include the Gelfond-Mahler inequality for the unit disk and the Kneser-Borwein inequality for the segment [−1, 1]. Furthermore, the asymptotically sharp constants are known for such inequalities over arbitrary compact sets in the complex plane. It is shown here that this best constant is smallest (namely: 2) for a disk. We also conjecture that it takes its largest value for a segment, among all compact connected sets in the plane. 1. The problem and its history Let E be a compact set in the complex plane C. For a function f: E → C define the uniform (sup) norm as follows: �f�E = sup |f(z)|. z∈E Clearly �f1f2 � E ≤ �f1 � E �f2 � E, but this inequality is not reversible, in general, not even with a constant factor in front of the right hand side. Indeed, �f1 � E �f2 � E ≤ C �f1f2 � E does not hold for functions with disjoint supports in E, for example. However, the situation is quite different for algebraic polyno-mials {pk(z)} m k=1 and their product p(z): = �m k=1 pk(z). Polynomial inequalities of the form m� (1.1) �pk�E ≤ C�p�E, k=1 exist and are readily available. One of the first results in this direction is due to Kneser [19], for E = [−1, 1] and m = 2 (see also Aumann [1]), who proved that (1.2) �p1�[−1,1]�p2�[−1,1] ≤ Kℓ,n�p1p2�[−1,1], deg p1 = ℓ, deg p2 = n − ℓ, 2000 Mathematics Subject Classification. Primary 30C10; Secondary 30C85, 31A15. Key words and phrases. Polynomials, products, factors, uniform norm, logarithmic capacity, equilibrium measure, subharmonic function, Fekete points. Research of I.P. was partially supported by the National Security Agency (grant H98230-06-1-0055), and by the Alexander von Humboldt Foundation. S.R. acknowledges partial support from the German-Israeli Foundation (grant G-809-234.6/2003), from FONDECYT (grants 1040366 and 7040069) and from DGIP-UTFSM (grant 240104). 1

Abstract. Expansive algebraic Z d-actions corresponding to ideals are characterized by the property that the complex variety of the ideal is disjoint from the multiplicative unit torus. For such actions it is known that the limit for the growth rate of periodic points exists and equals the entropy of the action. We extend this result to actions for which the complex variety intersects the multiplicative torus in a finite set. The main technical tool is the use of homoclinic points which decay rapidly enough to be summable. 1.

Dedicated to George G. Lorentz, whose works have been a great inspiration Abstract. We study the reverse triangle inequalities for suprema of logarithmic potentials on compact sets of the plane. This research is motivated by the inequalities for products of supremum norms of polynomials. We find sharp additive constants in the inequalities for potentials, and give applications of our results to the generalized polynomials. We also obtain sharp inequalities for products of norms of the weighted polynomials w n Pn, deg(Pn) ≤ n, and for sums of suprema of potentials with external fields. An important part of our work in the weighted case is a Riesz decomposition for the weighted farthest-point distance function.

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We already met a number of open problems in these notes, in particular in § 1.1.1. We collect further conjectures in this field, but this is only a very partial list of questions which deserve to be investigated further. Part of this section if from [W 2004], especially § 3. When K is a field and k a subfield, we denote by trdeg kK the transcendence degree of the extension K/k. In the case k = Q we write simply trdegK (see [La 1993] Chap. VIII, § 1).