Łojasiewicz exponents and Farey sequences

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 = ( φ 1 , φ 2 ) ∈ K [ [ t ] ] 2 , where φ ≠ ( 0 , 0 ) and φ ( 0 , 0 ) = ( 0 , 0 ) . The purpose of this paper is to investigate the invariant L 0 ( I ) = sup φ ∈ Φ inf f ∈ I ord f ∘ φ ord φ called the Łojasiewicz exponent of I . Our main result states that for the ideals I of finite codimension the Łojasiewicz 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 .