ZetaFunctionsNewtonND-Sagemath

A Sagemath class for computing (local) Igusa and topological zeta functions for Newton-non-degenerated polynomials

The file ZetaFunctionsNewtonND.py provides a class (the ZetaFunctions class) in Sagemath together with methods in order to study and compute local and global Igusa and topological zeta functions for Newton-non-degenerated polynomials.

This class is initialized with a multivariate polynomial and creates an object with several functions for computing the previous zeta functions whenever the associated polynomial is Newton-non-degenerated. Also, several methods to study the associated Newton polyhedron, its faces, dual cones, and possible candidate poles are provided.

This implementation is based in the formulas given in [DH01], [DL92] and [Var76] and an initial Maple program by K. Hoornaert and D. Loots in [HL00]. I would like to thank Frédéric Chapoton (Univ. Strasbourg), who improved outputs, handling errors and symbolic variables in a middle-stage version of this computer program.

Prerequisites

• Sagemath>=v9.4.

A minimal example

First, you should download ZetaFunctionsNewtonND.py and locate it in your working folder.

A minimal example could be:

sage: load('ZetaFunctionsNewtonND.py')
sage: R.<x,y,z,t> = QQ[]
sage: zex = ZetaFunctions(x^4 + y^3*z^3+y^3*t^3+z^3*t^3)
sage: zex.topological_zeta(local = True)
(1/4) * (s + 3/4)^-1 * (s + 11/12)^-1 * (s + 1)^-1 * (s^2 + 23/12*s + 11/4)
sage: zex.monodromy_zeta()
(t - 1)^-8 * (t + 1)^-8 * (t^2 + 1)^-17 * (t^2 - t + 1)^-9 * (t^2 + t + 1)^-9 * (t^4 - t^2 + 1)^-18

You can find much more examples and explanations in the Jupyter notebook Examples_ZetaFunctions.ipynb.

Methods in ZetaFunctions

Computing zeta functions

• igusa_zeta(self, p=None, dict_Ntau={}, local=False, weights=None, info=False, check='ideals')

• topological_zeta(self, d=1, local=False, weights=None, info=False, check='ideals')

• monodromy_zeta(self, char=False, info=False, check='ideals')

Newton polyhedron & cones: information and plotting

• cones_plot(self, **kwargs)

• is_newton_degenerated(self, p=None, local=False, method='default', info = False)

• get_newton_polyhedron(self)

• give_info_facets(self, compact = False)

• give_info_newton(self, faces=False, cones=False, compact = False)

• newton_plot(self, point_size = 30, **kwargs)

Information about (candidate) poles

• dict_info_poles(self, d=1, weights=None, local=False)

• get_polyfaces_dictionary(self, keys = 'polynomials', compact = False)

• give_expected_pole_info(self, d=1, local=False, weights=None)

References

[DH01] Denef, J. and Hoornaert, K., Newton Polyhedra and Igusa's Local Zeta Function, 2001. J. Number Theory 89 (2001), no. 1, 31–64.

[DL92] Denef, J. and Loeser, F., Caracteristiques d'Euler-Poincare, fonctions zeta locales et modifications analytiques, J. Amer. Math. Soc. 5 (1992), no. 4, 705–720.

[HL00] Hoornaert, K. and Loots, D., Computer program written in Maple for the calculation of Igusa's local zeta function, (2000).

[Var76] Varchenko, A. N., Zeta-function of monodromy and Newton's diagram. Invent. Math. 37 (1976), no. 3, 253–262.

[Viu12] Viu-Sos, J., Funciones zeta y poliedros de Newton: Aspectos teoricos y computacionales, Master Thesis, (2012).

Authors

• Kathleen Hoornaert (2000): initial version for Maple

• Juan Viu-Sos (2012): initial version for Sage

• Frédéric Chapoton (2017): improved outputs, handling errors and symbolic variables

• Juan Viu-Sos (2022): improved methods and new functions in the class to visualize Newton polyhedra

Contact

For any question or comment, please email me: juan.viu.sos@upm.es