熊宗洪,石昌梅,甘文良

(1.貴州民族大學(xué)理學(xué)院,貴州貴陽550025)

(2.貴州師范學(xué)院數(shù)學(xué)與計算機(jī)科學(xué)學(xué)院,貴州貴陽550018)

(3.東北師范大學(xué)數(shù)學(xué)與統(tǒng)計學(xué)院,吉林長春130024)

兩類二元函數(shù)芽的一個共同性質(zhì)及其應(yīng)用

熊宗洪1,石昌梅2,甘文良3

(1.貴州民族大學(xué)理學(xué)院,貴州貴陽550025)

(2.貴州師范學(xué)院數(shù)學(xué)與計算機(jī)科學(xué)學(xué)院,貴州貴陽550018)

(3.東北師范大學(xué)數(shù)學(xué)與統(tǒng)計學(xué)院,吉林長春130024)

本文主要研究二元C∞函數(shù)芽環(huán)中函數(shù)芽的性質(zhì)問題.利用Mather有限決定性定理和C∞函數(shù)的右等價關(guān)系,獲得了帶有任意4次至k次齊次多項式pi(x,y),qi(x,y)(i=4,5,···,k)的兩類函數(shù)芽的一個共同性質(zhì)：若M2k?M2J(fj)(j=1,2)且f1,f2的軌道切空間的余維分布均為ci=1(i=4,5,···,k-1),則對這里的i,pi(x,y)中xyi-1,yi的系數(shù)和qi(x,y)中xi-1y,xi的系數(shù)均為零.最后,利用該性質(zhì),給出了f1,f2和一類余維數(shù)為7的二元函數(shù)芽的標(biāo)準(zhǔn)形式.

二元函數(shù)芽;有限決定性;共同性質(zhì);標(biāo)準(zhǔn)形式;余維7

1 引言及準(zhǔn)備

設(shè)f∈M32?E2,則f的軌道切空間M2J(f)?M32,并考慮理想的序列套

En表示在O∈Rn處的C∞函數(shù)芽環(huán);Mn是En中的唯一極大理想;Mkn是Mn的k次冪;是k次齊次多項式全體構(gòu)成的實向量空間;jkf是f的k階Taylor多項式.

定義1.1設(shè)f,g∈En,若存在一個微分同胚φ∈Ln(為點O∈Rn處的局部微分同胚群),使得g=f?φ.則稱芽f與g是右等價的.

定義1.2[2]f∈En稱為有限k-決定的是指每一個與f有相同k階Taylor多項式的芽g是右等價于f的.

引理1.3[3](Nakayama引理)設(shè)I是En中的有限生成理想,則I等價于

引理1.5[4]設(shè)f(x)∈En,對于任意給定的局部微分同胚φ,則(f)等價于(證明思路見文[4]p.76-77).

由引理1.4,1.5,若f是k-決定的,則凡是與f右等價的函數(shù)芽也是k-決定的.

2 兩類函數(shù)芽的共同性質(zhì)

于是

3 兩類函數(shù)芽的應(yīng)用

由右等價關(guān)系的傳遞性知h(x,y)右等價于標(biāo)準(zhǔn)函數(shù)芽x2y±yk,即f1右等價于標(biāo)準(zhǔn)函數(shù)芽x2y±yk,其中”±”依賴于f1中yk項的系數(shù)符號.

定理3.2函數(shù)芽xy2±xk右等于x2y±yk.

證只需考慮線性同胚φ：R2→R2,(x,y)→(y,x)即可.

這樣一來,由定理2.1和定理3.1知f1和f2均右等價于標(biāo)準(zhǔn)函數(shù)芽x2y±yk.

定理3.3設(shè)f(x,y)∈M32是余維數(shù)為7的二元函數(shù)芽且其軌道切空間的余維分布為c3=c4=c5=1,c6=0,則f(x,y)的標(biāo)準(zhǔn)形式為x2y±y6,其中“±”依賴于f中y6項的系數(shù)符號.

[1]石昌梅,岑麗輝.余秩是2余維數(shù)為6的光滑實函數(shù)芽的分類[J].數(shù)學(xué)的實踐與認(rèn)識,2015,45(8)：253-260.

[2]Martinet J N.Singularities of smooth function and maps[M].London Math.Soc.Lec.Note Ser.58, Cambridge：Cambridge University Press,1982.

[3]Brocker T H.Dif f erentiable germs and catastrophes[M].London Math.Soc.Lec.Note Ser.17, Cambridge：Cambridge Press,1975.

[4]李養(yǎng)成.光滑映射的奇點理論[M].北京：科學(xué)出版社,2002.

[5]岑燕明.高階Morse芽的存在性[J].數(shù)學(xué)雜志,2006,26(3)：283-286.

A COMMON PROPERTY OF TWO TYPES OF FUNCTION GERMS WITH TWO VARIABLES AND THEIR APPLICATIONS

XIONG Zong-hong1,SHI Chang-mei2,GAN Wen-liang3
(1.School of Science,Guizhou Minzu University,Guiyang 550025,China)
(2.School of Mathematics and Computer Science,Guizhou Normal College,Guiyan 550018,China)
(3.School of Mathematics and Statistics,Northeast Normal University,Changchun 130024,China)

In this paper,we mainly consider a property of function germs in the ring of C∞functions germs of two variables.Using the Mather’s theorem of fi nite determinacy and right equivalence of functions,a common property of two types of function germskwith some arbitrary homogeneous polynomials pi(x,y)and qi(x,y)(i=4,5,···,k)of degree from 4 to k is obtained.(j=1,2)and the codimension distribution of tangent space of orbits for f1,f2are both ci=1(i=4,5,···,k-1), the coefficients of xyi-1and yiin pi(x,y)are both zero,so are the coefficients of xi-1y and xiin qi(x,y).Finally,by this property,the normal forms of f1,f2and a class of function germs of two variables with codimension 7 are given.

function germs of two variables; fi nite determinacy;common property;normal form;codimension 7

O186.16

A

0255-7797(2017)05-1087-06

2016-01-04接收日期：2016-03-28

貴州省科技廳聯(lián)合基金資助(黔科合LH字[2014]7378);貴州省數(shù)學(xué)建模及應(yīng)用創(chuàng)新人才團(tuán)隊項目基金資助(黔教科研發(fā)[2013]405號).

熊宗洪(1982-),男,苗族,貴州思南,講師,主要研究方向：奇點理論.

2010 MR Subject Classi fi cation：57R45