### D.15.2 arnold_lib

Library:
  arnold.lib
Purpose:
  Classification of isolated singularities with a nondegenerate Newton Boundary
Authors:
  Janko Boehm, email: boehm@mathematik.uni-kl.de
  Magdaleen Marais, email: msmarais@sun.ac.za
  Gerhard Pfister, email: pfister@mathematik.uni-kl.de

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.15.2.1 poshull |  | determine the convex hull of a list of monomials in mon[x,y]
> D.15.2.2 monomials |  | determine the set of monomials of a polynomial
> D.15.2.3 coeff |  | determine the coefficient of a given monomial of a polynomial
> D.15.2.4 newtonPolygon |  | determine the Newton polygon of the points corresponding to the monomials of a polynomial
> D.15.2.5 verticesOfNP |  | determine the vertices of a Newton polygon
> D.15.2.6 termsOnPolygon |  | determine the sum of the terms of a given polynomial of which the corresponding points lie on its corresponding Newton polygon
> D.15.2.7 latticePoints |  | determine all the points that lie on a given Newton polygon
> D.15.2.8 latticeToMonomials |  | determine the corresponding monomials of a list of points
> D.15.2.9 terms |  | determine the terms of a given polynomial
> D.15.2.10 piecewiseWeightOfPolygon |  | determine the piecewise weight defined by a given Newton polygon
> D.15.2.11 piecewiseOrd |  | determine the piecewise order of a given polynomial with regard to a given piecewise weight
> D.15.2.12 piecewisedegree |  | determine the piecewise degree of a given polynomial with regard to a given piecewise weight
> D.15.2.13 piecewiseJet |  | determine the piecewise jet of a given polynomial of a given degree, with regard to a given piecewise weight
> D.15.2.14 regularBasis |  | determine a regular basis for a polynomial with respect to the piecewise weight defined by its Newton polygon
> D.15.2.15 determineNormalForm |  | determine a normal form, modality, milnor number, delta invariant, number of branches, determinacy bound and corank of a given polynomial, if possible
> D.15.2.16 normalForm |  | determine a normalform for F.value, if F is of type Poly, or return F.normalForm
> D.15.2.17 determineExceptionalHypersurface |  | determine the exceptional hypersurface of the normalform stored in N.normalForm, and store the calculated hypersurface in the field N.exceptionalHypersurface
> D.15.2.18 exceptionalHypersurface |  | return N.exceptionalHypersurface
> D.15.2.19 determineNormalFormEquation |  | determine a normalform equation of a polynomial in the give normalform
> D.15.2.20 normalFormEquation |  | return N.normalFormEquation
> D.15.2.21 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.15.2.22 nondegeneratePart |  | determine the nondegenrate part of a singularity
> D.15.2.23 germWithNNB |  | determine a germ with a nondegenrate Newton boundary that is equivalent to a given polynomial, if possible
> D.15.2.24 determineGermWithSemiNormalizedNNB |  | determine a germ with a normalized nondegenerate Newton boundary that is equivalent to N.phi.sourcegerm.value
> D.15.2.25 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.15.2.26 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.15.2.27 milnorNNB |  | determine the Milnor number of a polynomial that is equivalent to a germ with a nondegenerate Newton boundary
> D.15.2.28 determinacyBound |  | determine an upper bound for the determinacy of a polynomial
> D.15.2.29 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.15.2.30 moduliMonomials |  | give the monomials corresponding to the moduli terms in a given normal form
> D.15.2.31 determineArnoldType |  | determine the Arnold classification of N.phi.sourcegerm and store it in the field N.ArnoldType
> D.15.2.32 ArnoldType |  | return the Arnold type of N.phi.sourcegerm.value
> D.15.2.33 newtonNumber |  | determine the Newton number of a polynomial in Q[x,y]
> D.15.2.34 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.15.2.35 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

See also: arnoldclassify_lib; classify2_lib; classify_lib; realclassify_lib.

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