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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 Phi denote the set of all parametrizations phi = (phi(1), phi(2)) is an element of K[[t]](2), where phi = (0, 0) and phi(0, 0) = (0, 0) . The purpose of this paper is to investigate the invariant

L-0(I) = sup(phi is an element of Phi) (inf(f is an element of I) ord f o phi/ord phi)

called 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.

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Łojasiewicz exponents and Farey sequences