D.15.1 arnold_lib

Library:

arnold.lib

Purpose:

Classification of isolated singularities with a nondegenerate Newton Boundary

Authors:

Janko Boehm, email: [email protected]
Magdaleen Marais, email: [email protected]
Gerhard Pfister, email: [email protected]

Overview:

We classify isolated singularities at 0 which is equivalent to a nondegenerate Newton boundary of corank <=2 with respect to right- and stable equivalence. We do this by
giving a unique normal form corresponding to a normalized nondegenerate Newton boundary. We furthermore determine the normal form equation, that is a polynomial in the
given normal form family of the input polynomial. In addition we determine the Milnor number, modality, delta invariant and the number of branches of an input polynomial
that is equivalent to a germ with a nondegenerate Newton boundary in a new alternative efficient way, as well as, a regular basis of a germ with a nondegenerate
Newton boundary.

V.I. Arnold has listed normal forms for all isolated hypersurface singularities over the complex numbers with, in particular, either modality <=2 or Milnor number <=16.
Moreover, he has described an algorithmic classifier, which determines the type of a given such singularity.
This library extends Arnold's work to a large class of singularities which is unbounded with regard to modality and Milnor number. It implements an algorithmic
classifier, which determines a normal form, as well as a normal form equation, for any corank <=2 singularity which is equivalent to a germ with non-degenerate Newton
boundary in the sense of Kouchnirenko.

The implementation is based on the papers:

Janko Boehm, Magdaleen Marais, Gerhard Pfister: Classification of Complex Singularities with Non-Degenerate Newton Boundary,
https://doi.org/10.48550/arXiv.2010.10185

Janko Boehm, Magdaleen Marais, Gerhard Pfister: Moduli Parameters of Complex Singularities with Non-Degenerate Newton Boundary,
https://doi.org/10.48550/arXiv.2402.05093

General assumption: the input polynomial is given in a ring with ordering ds;

Acknowledgements: This research was supported by the Rubbi fund of the Department of Mathematical Sciences of Stellenbosch University, DFG SPP 1489,
DFG TRR 195 (Project B5).

Procedures:

D.15.1.1 poshull		determine the convex hull of a list of monomials in mon[x,y]
D.15.1.2 monomials		determine the set of monomials of a polynomial
D.15.1.3 coeff		determine the coefficient of a given monomial of a polynomial
D.15.1.4 newtonPolygon		determine the Newton polygon of the points corresponding to the monomials of a polynomial
D.15.1.5 verticesOfNP		determine the vertices of a Newton polygon
D.15.1.6 termsOnPolygon		determine the sum of the terms of a given polynomial of which the corresponding points lie on its corresponding Newton polygon
D.15.1.7 latticePoints		determine all the points that lie on a given Newton polygon
D.15.1.8 latticeToMonomials		determine the corresponding monomials of a list of points
D.15.1.9 terms		determine the terms of a given polynomial
D.15.1.10 piecewiseWeightOfPolygon		determine the piecewise weight defined by a given Newton polygon
D.15.1.11 piecewiseOrd		determine the piecewise order of a given polynomial with regard to a given piecewise weight
D.15.1.12 piecewisedegree		determine the piecewise degree of a given polynomial with regard to a given piecewise weight
D.15.1.13 piecewiseJet		determine the piecewise jet of a given polynomial of a given degree, with regard to a given piecewise weight
D.15.1.14 regularBasis		determine a regular basis for a polynomial with respect to the piecewise weight defined by its Newton polygon
D.15.1.15 determineNormalForm		determine a normal form, modality, milnor number, delta invariant, number of branches, determinacy bound and corank of a given polynomial, if possible
D.15.1.16 normalForm		determine a normalform for F.value, if F is of type Poly, or return F.normalForm
D.15.1.17 determineExceptionalHypersurface		determine the exceptional hypersurface of the normalform stored in N.normalForm, and store the calculated hypersurface in the field N.exceptionalHypersurface
D.15.1.18 exceptionalHypersurface		return N.exceptionalHypersurface
D.15.1.19 determineNormalFormEquation		determine a normalform equation of a polynomial in the give normalform
D.15.1.20 normalFormEquation		return N.normalFormEquation
D.15.1.21 normalFormEquationUpToRescaling		N.normalFormEquationUpToRescaling, a germ which is right-equivalent to N.normalFormEquation, by the transformation x-->ax, y-->by, a,b complex numbers, such that its ring is minimal
D.15.1.22 nondegeneratePart		determine the nondegenrate part of a singularity
D.15.1.23 germWithNNB		determine a germ with a nondegenrate Newton boundary that is equivalent to a given polynomial, if possible
D.15.1.24 determineGermWithSemiNormalizedNNB		determine a germ with a normalized nondegenerate Newton boundary that is equivalent to N.phi.sourcegerm.value
D.15.1.25 germWithSemiNormalizedNNB		return a germ with a nondegenerate normalized Newton boundary, up to scalar multiplication of each of its variables, of a given polynomial, if possibles
D.15.1.26 modalityNNB		determine the modality of the singularity defined by a polynomial,if the polynomial is right equivalent to a germ with a nondegenerate Newton boundary
D.15.1.27 milnorNNB		determine the Milnor number of a polynomial that is equivalent to a germ with a nondegenerate Newton boundary
D.15.1.28 determinacyBound		determine an upper bound for the determinacy of a polynomial
D.15.1.29 deltaNNB		determine the Delta invariant and number of branches of a polynomial that is right equivalent to a germ with a nondegenerate Newton boundary
D.15.1.30 moduliMonomials		give the monomials corresponding to the moduli terms in a given normal form
D.15.1.31 determineArnoldType		determine the Arnold classification of N.phi.sourcegerm and store it in the field N.ArnoldType
D.15.1.32 ArnoldType		return the Arnold type of N.phi.sourcegerm.value
D.15.1.33 newtonNumber		determine the Newton number of a polynomial in Q[x,y]
D.15.1.34 transformationsBeforeSplit		the transformations (and in some cases their inverses) that was transformed on a given polynomial to write it as a direct sum of its degenerate and nondegenerate parts, given up to filtration d, where d is a determinacy bound for the given polynomial
D.15.1.35 transformationsAfterSplit		the transformations (and in some cases their inverses) that was transformed on the degenerate part of a given polynomial was splitted off to transform it to a germ with nondegenerate Newton boundary (in some cases the transformations normalize the Newton boundary is also given), is given up to filtration d, where d is a determinacy bound for the polynomial, if possible