Simultaneous diophantine approximation for algebraic functions in positive characteristic

Abstract

Abstract. We study simultaneous rational approximations of a pair of algebraic functions in positive characteristic. We particularly consider quadratic functions in characteristic 2.

Simultaneous Diophantine Approximation in Function Fields

“…For a general background to material on Diophantine approximation in characteristic zero and in positive characteristic, the reader should consult [5,6,7,24]. Certainly, there are fruitful results on Diophantine approximation in positive characteristic (see the survey papers [13,25]), but there is very little information about simultaneous approximation (see [10,12,16,18]). The first significant progress on high dimensional approximation is due to K. Mahler [16], who established an analogue to Minkowski's second theorem on the geometry of numbers in positive characteristic (see also Chapter 9 of [6]).…”

“…The key point of connection is Lemma 2.4 below, known as Minkowski's linear forms theorem (see Theorem III of Appendix B of [4]). In [18], deMathan only dealt with a pair of algebraic functions approximated by rational functions, in other words, he only obtained a special two dimensional result in the homogeneous case.…”

“…The approximation constant τ (θ) given by (1.12) is a standard notion in real number fields (see page 11 of [4]). According to [13,18,19], we may generalize this definition as follows: Let θ 1 , . .…”

On the Littlewood conjecture in fields of power series

“…In this Section, we discuss whether the (function field analogue of the) Littlewood conjecture holds for pairs of algebraic power series defined over a finite field k. Our knowledge is slightly better than in the real case, especially thanks to works of Baum and Sweet [4] and of de Mathan [19,20,21,22] that we recall below.…”

“…On the other hand, there are several results on pairs of algebraic functions that satisfy non-trivially the Littlewood conjecture. De Mathan [21] established that (1.2) holds for any pair (Θ, Φ) of quadratic elements when k is any finite field of characteristic 2 (see also [19,20] for results when k is any finite field). Furthermore, he proved in [22] the analogue of the Cassels and Swinnerton-Dyer theorem when k is a finite field.…”