Quantcast

Łojasiewicz exponents and Farey sequences

Research paper by A. B. de Felipe, E. R. García Barroso; J. Gwoździewicz; A. P�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 \(0<b<a<N\) \(0<b<a<N\)