Diophantine approximation of polynomials over 𝔽q[t] satisfying a divisibility condition

2016 ◽  
Vol 12 (05) ◽  
pp. 1371-1390 ◽  
Author(s):  
Shuntaro Yamagishi
Keyword(s):  
Finite Field ◽  
Diophantine Approximation ◽  
Linear Combinations ◽  
Ring Of Polynomials

Let [Formula: see text] denote the ring of polynomials over [Formula: see text], the finite field of [Formula: see text] elements. We prove an estimate for fractional parts of polynomials over [Formula: see text] satisfying a certain divisibility condition analogous to that of intersective polynomials in the case of integers. We then extend our result to consider linear combinations of such polynomials as well.

scholarly journals Linear combinations of primitive elements of a finite field

2018 ◽  
Vol 51 ◽  
pp. 388-406 ◽  
Author(s):  
Stephen D. Cohen ◽  
Tomás Oliveira e Silva ◽  
Nicole Sutherland ◽  
Tim Trudgian
Keyword(s):  
Finite Field ◽  
Primitive Elements ◽  
Linear Combinations

LOGICAL CHARACTERIZATION OF RECOGNIZABLE SETS OF POLYNOMIALS OVER A FINITE FIELD

2011 ◽  
Vol 22 (07) ◽  
pp. 1549-1563 ◽  
Author(s):  
MICHEL RIGO ◽  
LAURENT WAXWEILER
Keyword(s):  
Finite Field ◽  
Order Theory ◽  
Ring Of Integers ◽  
First Order ◽  
Polynomial Coefficients ◽  
Bounded Number ◽  
First Order Theory ◽  
Ring Of Polynomials

The ring of integers and the ring of polynomials over a finite field share a lot of properties. Using a bounded number of polynomial coefficients, any polynomial can be decomposed as a linear combination of powers of a non-constant polynomial P playing the role of the base of the numeration. Having in mind the theorem of Cobham from 1969 about recognizable sets of integers, it is natural to study P-recognizable sets of polynomials. Based on the results obtained in the Ph.D. thesis of the second author, we study the logical characterization of such sets and related properties like decidability of the corresponding first-order theory.

scholarly journals Continued fractions and diophantine equations in positive characteristic

2021 ◽  
Vol 109 (123) ◽  
pp. 143-151
Author(s):  
Khalil Ayadi ◽  
Awatef Azaza ◽  
Salah Beldi
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Diophantine Equations ◽  
Positive Characteristic ◽  
Continued Fraction Expansion ◽  
Ring Of Polynomials ◽  
Algebraic Power Series

We exhibit explicitly the continued fraction expansion of some algebraic power series over a finite field. We also discuss some Diophantine equations on the ring of polynomials, which are intimately related to these power series.

scholarly journals Number systems and tilings over Laurent series

2009 ◽  
Vol 147 (1) ◽  
pp. 9-29 ◽  
Author(s):  
TOBIAS BECK ◽  
HORST BRUNOTTE ◽  
KLAUS SCHEICHER ◽  
JÖRG M. THUSWALDNER
Keyword(s):  
Finite Field ◽  
Fundamental Domain ◽  
Laurent Series ◽  
Residue Class ◽  
Number Systems ◽  
Fundamental Domains ◽  
Ring Of Polynomials ◽  
Residue Class Ring ◽  
Class Ring

AbstractLet be a field and [x, y] the ring of polynomials in two variables over . Let f ∈ [x, y] and consider the residue class ring R := [x, y]/f[x, y]. Our first aim is to study digit representations in R, i.e., we ask for which f each element of R admits a digit representation of the form d0 + d1x + ⋅ ⋅ ⋅ + dℓxℓ with digits di ∈ [y] satisfying degy(di) < degy(f). These digit systems are motivated by the well-known notion of canonical number systems. Next we enlarge the ring in order to allow representations including negative powers of the “base” x. In particular, we define and characterize digit representations for the ring S := ((x−1, y−1))/f((x−1, y−1)) and give easy to handle criteria for finiteness and periodicity of such representations. Finally, we attach fundamental domains to our digit systems. The fundamental domain of a digit system is the set of all elements having only negative powers of x in their “x-ary” representation. The translates of the fundamental domain induce a tiling of S. Interestingly, the fundamental domains of our digit systems turn out to be unions of boxes. If we choose =q to be a finite field, these unions become finite.

Smooth Polynomial Solutions to a Ternary Additive Equation

2018 ◽  
Vol 70 (1) ◽  
pp. 117-141 ◽  
Author(s):  
Junsoo Ha
Keyword(s):  
Finite Field ◽  
Large Integer ◽  
Polynomial Solutions ◽  
Ring Of Polynomials ◽  
Additive Equation

AbstractLet Fq[T] be the ring of polynomials over the finite field of q elements and Y a large integer. We say a polynomial in Fq[T] is Y-smooth if all of its irreducible factors are of degree at most Y. We show that a ternary additive equation a + b = c over Y-smooth polynomials has many solutions. As an application, if S is the set of first s primes in Fq[T] and s is large, we prove that the S-unit equation u + v = 1 has at least exp solutions.

Diophantine Approximation for Certain Algebraic Formal Power Series in Positive Characteristic

2013 ◽  
Vol 56 (4) ◽  
pp. 673-683
Author(s):  
K. Ayadi ◽  
M. Hbaib ◽  
F. Mahjoub
Keyword(s):  
Power Series ◽  
Finite Field ◽  
Formal Power Series ◽  
Diophantine Approximation ◽  
Positive Characteristic ◽  
Rational Approximations ◽  
Positive Degree ◽  
Formal Power ◽  
Algebraic Power Series

Abstract.In this paper, we study rational approximations for certain algebraic power series over a finite field. We obtain results for irrational elements of strictly positive degree satisfying an equation of the typewhere (A, B, C) ∊ (𝔽q[X])2 × 𝔽*q [X]. In particular, under some conditions on the polynomials A, B and C, we will give well approximated elements satisfying this equation.

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