ReducedFiniteDimensionalModules

Code accompanying the paper: "Reduced submodules of finite dimensional polynomial modules". Any object described or referred to here is defined and studied in the paper.

All to be run in python3

Purpose

The purpose of this code is the aid of the study of reduced submodules of finite dimensional polynomial modules. Let k be a field, R = k[x,y] and I be a monomial ideal of R. Consider the module M = R/I. We wish to study the subset R(M) of M called the maximal reduced submodule. A monomial ideal of R can be given by a Young diagram.

"main.py" takes as input a Young diagram and outputs generators for the corresponding module M and for R(M). It also outputs which of the four possible types R(M) is.

"data.py" file cycles through every possible module up to a given dimension and records the distribution of the four types, saving the data.

The pictures.py file produces pictures of the Young diagrams associated with each of the modules and a picture which highlights the differences between M and R(M). This is to help researchers consider many pictorial examples in order to gain quickly intuition.

Usage

main.py

Input taken as standard input as a decreasing sequence of positive integers separated by a space. For example: ``` Enter Young diagram in the form: * * ... *

10 9 3 3 2 1 Output is given is the following form: Generators of M : {(3, -2), (0, -6), (1, -5), (10, 0), (2, -4), (9, -1)} R(M) generators : {(1, -4), (9, 0), (8, -1), (3, -2), (2, -3), (0, -5)} Type : [0, 0, 1, 0] ``` Where a pair (a,-b) represents a monomial $x^ay^b$ and a 1 in the $i^{th}$ position means type $i$ for $i = 1,...,4$.

In order to exit the program type: exit

data.py

Input is given as maximum dimension to be tested. Then one enters the name of the file where the data will be saved. For example: Enter the maximum dimension: 5 Name file to save data under: data-test In the file then is saved a file called "data-test.txt". It will have the following form: ``` Dim T1 T2 T3 T4 2 100.0 0.0 0.0 0.0 3 66.67 33.33 0.0 0.0 4 40.0 40.0 20.0 0.0 5 28.57 28.57 42.86 0.0 ~

"data-test.txt" [noeol] 5L, 244C ``` In each collumn is the percentage share of that type of the given dimension.

pictures.py

Input taken as standard input as a decreasing sequence of positive integers separated by a space. For example: ``` Enter Young diagram in the form: * * ... *

10 9 3 3 2 1 Output is of the following form:

Type = [0, 0, 1, 0]

||||||||||
|||||||||
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_|||
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|||||||||*| _||||||||| ||_| ||| _|| *|

              |*|
            |*|

|*|

|| || The first diagram is the original young diagram. The second is with the generators of R(M) marked by*``` and the third shows the overlap.

In order to exit the program type: exit

References

Reduced submodules of finite dimensional polynomial modules - Tilahun Abebaw, Nega Arega, Teklemichael Worku Bihonegn, Dominic Bunnett, David Ssevviiri

Authors

Contact Dominic Bunnett in case of questions.