Łojasiewicz exponents and Farey sequences

Abstract

Let I be an ideal of the ring of formal power series K[ [x, y]] with coefficients in an algebraically closed field K of arbitrary characteristic. Let Φ denote the set of all parametrizations ϕ = (where ϕ = (0, 0) and ϕ(0, 0) = (0, 0). The purpose of this paper is to investigate the invariantcalled the Lojasiewicz exponent of I. Our main result states that for the ideals I of finite codimension the Lojasiewicz exponent L 0 (I) is a Farey number i.e. an integer or a rational number of the form N + b a , where a, b, N are integers such that 0 < b < a < N .

The Łojasiewicz exponent via the valuative Hamburger-Noether process

“…The case of ideals in k[[x, y]], where k is as above, is due to the authors [3]. De Felipe, García Barroso, Gwoździewicz and Płoski [7] gave a shorter proof of this result; moreover, they answered [3,Question 1], by showing that L(a) is always a Farey number, i. e. a rational number of the form N + b/a, where N , a, b are integers such that 0 < b < a < N .…”