Abstract

A complete classification of simple function germs with respect to Lipschitz equivalence over the field of complex numbers was given by Nguyen et al. The aim of this article is to implement a classifier in terms of easy computable invariants to compute the type of the Lipschitz simple function germs without computing the normal form in the computer algebra system Singular.

1. Introduction and Preliminaries

Let be a field and be the collection of all germs of smooth function at . The collection of all germs such that is denoted by . If , then it is equivalent to consider as , the local ring of formal power series in variables and its maximal ideal. Let , the set of all -automorphisms of , and , the set of all -bi-Lipschitz homeomorphisms of . Two smooth germs and are right equivalent (resp., bi-Lipschitz right equivalent) denoted by (resp., ) if there exists an automorphism (resp., a bi-Lipschitz homeomorphisms ) such that . In case of two variables (resp., three variables), we will later use (resp., ) instead of (resp., ) [13].

In seventies, Arnold [1–3] introduced the notion of modality for singularities over the fields and . He gave a classification of hypersurfaces of modality , and 2 under right equivalence. These are also classifications with respect to contact equivalence. Also we have the contributions by Guisti [4], Wall [5], and many others [6–11]. Mostowski [12] showed that germs of complex analytic set do not admit moduli with respect to bi-Lipschitz equivalence. Note that the result of Mostowski does not hold for function germs. Henry and Parusiński [13] proved that function germs do admit moduli under bi-Lipschitz right equivalence. Nguyen et al. [14] gave the classification of Lipschitz simple function germs. The aim of this article is to implement a classifier for this classification in the computer algebra system SINGULAR [15].

Let ; then -jet of denoted by is the Taylor expansion of up to degree terms. Let for all and , then is called -determined. A finitely determined germ is Lipschitz 0-modal if there is a neighborhood , the jet of for sufficiently large that meets only finitely many bi-Lipschitz equivalence classes. We use the following invariants for our classifier.

Definition 1. The Taylor expansion of at 0 is an expression of the formwhere each is a homogeneous polynomial of degree and . In this expansion, denotes the homogenous polynomial of lowest degree, and we denote it by . Then is the multiplicity of at 0.

Definition 2. Let be a Jacobian ideal and be the Milnor algebra over algebra. Then is called the codimension of .

Definition 3. Let and be its corresponding Hessian matrix. Then rank is called the corank of and is denoted by corank.

Remark 1. Lemma 4.2, Lemma 4.3, and Theorem 4.7 of [14] give the bi-Lipschitz invariants , , and corank.

2. A Classifier for Simple Function Germs under Lipschitz Equivalence

In this section, we present propositions and algorithms deduced from propositions to characterize the simple functions germs with respect to Lipschitz equivalence in terms of certain invariants such as multiplicity, corank, and codimension of . To differentiate some cases, we use locus of . For a proof of the following results, see Theorem 8.4, Theorem 8.5, and Theorem 8.7 of [14].

Theorem 1 (see [14]). A germ is Lipschitz simple if and only if it is bi-Lipschitz equivalent to one of the normal forms given in Tables 1 and 2.

Table 1 
Lipschitz Simple Function Germs of corank 1 and 2.
Table 2 
Lipschitz simple function germs of corank 3.

Theorem 2 (see [14]). Every germ with corank greater or equal to 4 is the Lipschitz modal.

2.1. Lipschitz Simple Function Germs of Corank 1 and 2

In the following, Propositions 1 and 2 give the possible overlapping of Lipschitz simple germs in case of corank 2.

Proposition 1. Let be a map germ of corank 2 and multiplicity 3.(1)If codimension ofis 6, thenis Lipschitz simple of typeor.(2)If codimension ofis 7, thenis Lipschitz simple of typeor.(3)If codimension ofis 8, thenis Lipschitz simple of typeor.

Proposition 2. Let be a map germ of corank 2 and multiplicity 4. If codim, then is Lipschitz simple of type or .

Proof. The statements of Propositions 1 and 2 follow Theorem 8.4 of [14], and these overlappings can be differentiated by computing the zero set of .

 Input:.
 Output: Type of w.r.t. Lipschitz equivalence.
(1)Compute , the corank of ;
(2)Compute , the codimension of ;
(3)Compute , the multiplicity of ;
(4)if, then
(5) return is of type ;
(6)end if
(7)if and , then
(8)  Compute , the zero set of the lowest degree homogeneous part of ;
(9)  if is the intersection of a line and a double line, then
(10)   return is of type ;
(11)end if
(12)  if is a triple line and , or 8, then
(13)   return is of type ;
(14)end if
(15)end if
(16)  if and , then
(17) if, then
(18)   return is of type ;
(19)end if
(20)  if, then
(21)   return is of type ;
(22)end if
(23)  if, then
(24)  Compute , the zero set of the lowest degree homogeneous part of ;
(25)  if is the intersection of a line and a triple line, then
(26)   return is of type ;
(27)end if
(28)if is the intersection of two double lines, then
(29)   return is of type ;
(30)end if
(31)end if
(32) if, then
(33)  return is of type ;
(34)end if
(35)end if
Algorithm 1 
Type of , when corank of or 2.
 Input:.
 Output: Type of w.r.t. Lipschitz equivalence.
(1)Compute , the corank of ;
(2)Compute , the codimension of ;
(3)Compute , the multiplicity of ;
(4)if, then
(5) return is of type ;
(6)end if
(7)if, then
(8)return is of type ;
(9)end if
(10)if, then
(11)  Compute , the zero set of the lowest degree homogeneous part of ;
(12)  if is irreducible and has in the singular locus a fat point of multiplicity 6, then
(13)   return is of type ;
(14)end if
(15)  if is irreducible and has in the singular locus a fat point of multiplicity 4, then
(16)   return is of type ;
(17)end if
(18)end if
(19)if, then
(20)  Compute , the zero set of the lowest degree homogeneous part of ;
(21)  if is the intersection of a plane and a node and has as a singular locus, the union of the lines and the point as embedded point, then
(22)   return is of type ;
(23)end if
(24)  if is the intersection of 3 planes and has as singular locus, the union of three lines , then
(25)   return is of type ;
(26)end if
(27)  if is irreducible and has in the singular locus a fat point of multiplicity 4, then
(28)   return is of type ;
(29)end if
(30)  if is the intersection of a plane and a node and has as a singular locus, the line , then
(31)   return is of type ;
(32)end if
(33)end if
Algorithm 2 
Type of , when corank of .
(1)if, then
(2)  Compute , the zero set of the lowest degree homogeneous part of ;
(3)  if is the intersection of a plane and a node and has as a singular locus, the union of the lines and the point as embedded point, then
(4)return is of type ;
(5)end if
(6)if is the intersection of 3 planes and has as singular locus, the union of three lines , then
(7)   return is of type ;
(8)end if
(9)  if is the intersection of a plane and a node and has as a singular locus, the line , then
(10)   return is of type ;
(11)end if
(12)end if
(13)  if, then
(14)   return is of type ;
(15)end if
(16)  if, then
(17)   return is of type ;
(18)end if
Algorithm 3 
Continuation of Algorithm 2.
2.2. Lipschitz Simple Function Germs of Corank 3

The following Proposition 3 describes the possible overlappings of Lipschitz simple germs in case of corank 3.

Proposition 3. Let be a map germ of corank 3 and multiplicity 3.(1)If codimension ofis 10, thenis Lipschitz simple of typeor.(2)If codimension ofis 11, thenis Lipschitz simple of type,,, or.(3)If codimension ofis 12, thenis Lipschitz simple of type,, or.

Proof. The statement follows from Theorem 8.5 of [14], and the overlappings can be differentiated by computing the zero set of the .

2.3. Singular Examples

We have implemented the Algorithm in the computer algebra system SINGULAR [15]. Code can be downloaded from https://www.mathcity.org/files/ahsan/classiFyLip-Procrdure.txt.

We give some examples: ring R=0, (x, y), ds;  poly f=x4−x3y−3x2y2+5xy3−2y4+2x5+17x4y+59x3y2+88x2y3+55xy4+22y5     +2x4y2+5x3y3+30x2y4+34xy5+10y6+9x5y2+60x4y3+147x3y4+138x2y5     +52xy6+2y7+4x4y4+24x3y5+51x2y6+29xy7+16x5y4+78x4y5+120x3y6     +56x2y7+xy8+6x4y6+24x3y7+24x2y8+14x5y6+44x4y7+32x3y8+4x4y8     +8x3y9+6x5y8+9x4y9+x4y10+x5y10; > LclassiFy2(f); f is of type Z11  poly g=x3−3xy2+2y3+2x2y2+2xy3−4y4+x3y2−2x2y3+2xy4+2y5+2x2y4−2xy5     +x7+14x6y+84x5y2+280x4y3+560x3y4+672x2y5+449xy6+128y7+7x7y2     +84x6y3+420x5y4+1120x4y5+1680x3y6+1344x2y7+448xy8+21x7y4     +210x6y5+840x5y6+1680x4y7+1680x3y8+672x2y9+35x7y6+280x6y7     +840x5y8+1120x4y9+560x3y10+35x7y8+210x6y9+420x5y10+280x4y11     +21x7y10+84x6y11+84x5y12+7x7y12+14x6y13+x7y14; > LclassiFy2(g); f is of type D8 ring R=0,(x,y,z),ds;  poly h=2x3+5x2y+4xy2+y3+5x2z+9xyz+4y2z+4xz2+4yz2+z3+x4+4x3z+6x2z2     +4xz3+z4+x5+5x4y+10x3y2+10x2y3+5xy4+y5; > LclassiFy3(h); f is of type T_3, 4, 5  poly p=2x3+3x2y+xy2+5x2z+6xyz+y2z+4xz2+3yz2+z3+x4+4x3y+6x2y2+4xy3     +y4+x3z+3x2yz+3xy2z+y3z+x5+5x4y+10x3y2+10x2y3+5xy4+y5; > LclassiFy3(p); f is of type S12

3. Conclusion

The aim of this paper is to implement the classification of simple function germs with respect to the Lipschitz equivalence given by Nguyen et al. in computer algebra system Singular. The proposed algorithms compute the type of the Lipschitz simple function germs without computing its normal form.

Data Availability

The data used to support the findings of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This work was supported by the Fund for Excellent Young Talents in Universities of Anhui Province (Research on Decentralized E-Commerce Credit System Model Construction and Optimization Strategy Based on Blockchain Technology. A Case Study of Anhui Province (Project No.: gxyq2018251)).