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Effective bounds for the zeros of linear recurrences in function fields

Clemens FuchsAttila Pethő — 2005

Journal de Théorie des Nombres de Bordeaux

In this paper, we use the generalisation of Mason’s inequality due to Brownawell and Masser (cf. [8]) to prove effective upper bounds for the zeros of a linear recurring sequence defined over a field of functions in one variable. Moreover, we study similar problems in this context as the equation G n ( x ) = G m ( P ( x ) ) , ( m , n ) 2 , where ( G n ( x ) ) is a linear recurring sequence of polynomials and P ( x ) is a fixed polynomial. This problem was studied earlier in [14,15,16,17,32].

Substitutions, abstract number systems and the space filling property

Clemens FuchsRobert Tijdeman — 2006

Annales de l’institut Fourier

In this paper we study multi-dimensional words generated by fixed points of substitutions by projecting the integer points on the corresponding broken halfline. We show for a large class of substitutions that the resulting word is the restriction of a linear function modulo 1 and that it can be decided whether the resulting word is space filling or not. The proof uses lattices and the abstract number system associated with the substitution.

Composite rational functions expressible with few terms

Clemens FuchsUmberto Zannier — 2012

Journal of the European Mathematical Society

We consider a rational function f which is ‘lacunary’ in the sense that it can be expressed as the ratio of two polynomials (not necessarily coprime) having each at most a given number of terms. Then we look at the possible decompositions f ( x ) = g ( h ( x ) ) , where g , h are rational functions of degree larger than 1. We prove that, apart from certain exceptional cases which we completely describe, the degree of g is bounded only in terms of (and we provide explicit bounds). This supports and quantifies the intuitive...

Commutative algebraic groups and p-adic linear forms

Clemens FuchsDuc Hiep Pham — 2015

Acta Arithmetica

Let G be a commutative algebraic group defined over a number field K that is disjoint over K from a and satisfies the condition of semistability. Consider a linear form l on the Lie algebra of G with algebraic coefficients and an algebraic point u in a p-adic neighbourhood of the origin with the condition that l does not vanish at u. We give a lower bound for the p-adic absolute value of l(u) which depends up to an effectively computable constant only on the height of the linear form, the height...

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