Examples

All examples are for \(K=\mathbb Q\) equipped with the 2-adic valuation.

Random lexicographical Groebner bases of ideals in shape position

A random lexicographical Groebner basis \(G\) of an ideal \(I\subseteq \mathbb Q[x_1,\ldots,x_n]\) in shape position is of the form

\[\{ f_n(x_n), x_{n-1} - f_{n-1}(x_n), \ldots, x_1-f_1(x_n) \},\]

where

  • \(f_n,f_{n-1},\ldots,f_1\) are univariate polynomials of degrees \(d,d-1,\ldots,d-1\) respectively,

  • all coefficients are of the form \(2^\lambda \cdot (2k+1)\) for some random valuation \(0\leq\lambda < 100\) and some random \(0\leq k < 5000\).

In our paper, we have constructed 100 random lexicographical Groebner bases for \(d=2,4,8,12,16,20,24\). You can load all examples into Singular by running the script as follows:

                     SINGULAR                                /  Development
A Computer Algebra System for Polynomial Computations       /   version 4.1.1
                                                         0<
by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Feb 2018
FB Mathematik der Universitaet, D-67653 Kaiserslautern       \
> <"lexicographicalGroebnerBases.sing".

Differentiated by degree, the timings are:

../_images/timingShape.sing

27 distinct tropical lines on a random honeycomb cubic

In [PV19], it is shown that a general tropical cubic surface \(\text{Trop}(f)\) in \(\mathbb T\mathbb P^3\) in honeycomb form contains 27 distinct tropical lines, which must then be tropicalizations of the 27 lines on any algebraic cubic \(\text{V}(f)\) in \(\mathbb P^3\).

In our paper, we have constructed 1000 random cubic polynomials, whose coefficients are pure powers of 2, and computed the lines on the tropical cubic surfaces as points in \(\text{Grass}(2,4)\) in the affine chart where Pluecker coordinate \(p_{34}\) is 1.

You can load all examples into Singular by running the script as follows:

                     SINGULAR                                /  Development
A Computer Algebra System for Polynomial Computations       /   version 4.1.1
                                                         0<
by: W. Decker, G.-M. Greuel, G. Pfister, H. Schoenemann     \   Feb 2018
FB Mathematik der Universitaet, D-67653 Kaiserslautern       \
> <"timingExamplesCubics.sing".

Differentiated by splitting field, the timings are:

../_images/timingCubics.sing