# Łojasiewicz exponents and Farey sequences

Research paper by A. B. de Felipe, E. R. GarcÃ­a Barroso; J. GwoÅºdziewicz; A. PÅ�oski

Indexed on: 24 Aug '16Published on: 01 Sep '16Published in: Revista Matemática Complutense

#### Abstract

Let I be an ideal of the ring of formal power series $$\mathbf {K}[[x,y]]$$ with coefficients in an algebraically closed field $$\mathbf {K}$$ of arbitrary characteristic. Let $$\varPhi$$ denote the set of all parametrizations $$\varphi =(\varphi _1,\varphi _2)\in \mathbf {K}[[t]]^2$$ , where $$\varphi \ne (0,0)$$ and $$\varphi (0,0)=(0,0)$$ . The purpose of this paper is to investigate the invariant \begin{aligned} \mathcal{{L}}_0(I)=\sup _{\varphi \in \varPhi }\left( \inf _{f\in I} \frac{\mathop {{{\mathrm {ord}}}}f \circ \varphi }{\mathop {{{\mathrm {ord}}}}\varphi }\right) \end{aligned} called the Łojasiewicz exponent of I. Our main result states that for the ideals I of finite codimension the Łojasiewicz exponent $$\mathcal{{L}}_0(I)$$ is a Farey number i.e. an integer or a rational number of the form $$N+\frac{b}{a}$$ , where a, b, N are integers such that $$0<b<a<N$$ . Let I be an ideal of the ring of formal power series $$\mathbf {K}[[x,y]]$$ with coefficients in an algebraically closed field $$\mathbf {K}$$ of arbitrary characteristic. Let $$\varPhi$$ denote the set of all parametrizations $$\varphi =(\varphi _1,\varphi _2)\in \mathbf {K}[[t]]^2$$ , where $$\varphi \ne (0,0)$$ and $$\varphi (0,0)=(0,0)$$ . The purpose of this paper is to investigate the invariant \begin{aligned} \mathcal{{L}}_0(I)=\sup _{\varphi \in \varPhi }\left( \inf _{f\in I} \frac{\mathop {{{\mathrm {ord}}}}f \circ \varphi }{\mathop {{{\mathrm {ord}}}}\varphi }\right) \end{aligned} called the Łojasiewicz exponent of I. Our main result states that for the ideals I of finite codimension the Łojasiewicz exponent $$\mathcal{{L}}_0(I)$$ is a Farey number i.e. an integer or a rational number of the form $$N+\frac{b}{a}$$ , where a, b, N are integers such that $$0<b<a<N$$ .I $$\mathbf {K}[[x,y]]$$ $$\mathbf {K}[[x,y]]$$ $$\mathbf {K}$$ $$\mathbf {K}$$ $$\varPhi$$ $$\varPhi$$ $$\varphi =(\varphi _1,\varphi _2)\in \mathbf {K}[[t]]^2$$ $$\varphi =(\varphi _1,\varphi _2)\in \mathbf {K}[[t]]^2$$ $$\varphi \ne (0,0)$$ $$\varphi \ne (0,0)$$ $$\varphi (0,0)=(0,0)$$ $$\varphi (0,0)=(0,0)$$ \begin{aligned} \mathcal{{L}}_0(I)=\sup _{\varphi \in \varPhi }\left( \inf _{f\in I} \frac{\mathop {{{\mathrm {ord}}}}f \circ \varphi }{\mathop {{{\mathrm {ord}}}}\varphi }\right) \end{aligned} \begin{aligned} \mathcal{{L}}_0(I)=\sup _{\varphi \in \varPhi }\left( \inf _{f\in I} \frac{\mathop {{{\mathrm {ord}}}}f \circ \varphi }{\mathop {{{\mathrm {ord}}}}\varphi }\right) \end{aligned}Łojasiewicz exponentII $$\mathcal{{L}}_0(I)$$ $$\mathcal{{L}}_0(I)$$ $$N+\frac{b}{a}$$ $$N+\frac{b}{a}$$abN $$0<b<a<N$$ $$0<b<a<N$$