Gröbner Valuations

The Macaulay2 package GrobnerValuations allows the user to compute Gröbner bases over the rational numbers equipped with the p-adic valuation. It contains algorithms from the paper "Gröbner bases over fields with valuations" by Andrew J Chan and Diane Maclagan which is available on the arXiv:1303.0729.

In order to use this package you need Macaulay2 installed on your computer. It is available as a free download.

The key commands are grobnerVal which computes the Gröbner basis over the rationals with the p-adic valuation, and initForm which computes the initial ideal.

Basic Instructions for Use

  1. Save the file as GrobnerValuations.m2 in a directory from which you can start Macaulay 2.
  2. Start Macaulay 2, and then install the package by entering installPackage "GrobnerValuations".
  3. Entering viewHelp GrobnerValuations will open a browser window with the documentation giving more information on how to use the package.
  4. As an example of how to use grobnerVal, you first need to specify a polynomial ring, by entering for example QQ[x,y,z]; followed by an ideal, by entering for example I = ideal(7*x^4*z-18*y^5,2*x^3*z^4-3*y^7); before using grobnerVal by entering for example grobnerVal(I,{2,1,4},Prime=>3) which computes the Gröbner basis for the ideal I with respect to the weight vector (2,1,4) in the polynomial ring with coefficients the rationals with the 3-adic valuation.
  5. initForm can be used similarly to grobnerVal by again specifying the ideal then entering for example initForm(I,{2,1,4},Prime=>3). initForm can also be used to compute the initial form of polynomials by entering for example initForm(7*x^4*z-18*y^5,{2,1,4},Prime=>3).