Decomposing Univariate Polynomials over Finite Fields

This Python-module provides functions to study the functional decomposition of univariate polynomials over a finite field. We have the following fundamental distinctions.

Tame case

where the polynomial's degree is coprime to the characteristic of the base field.

Wild case

where the polynomial's degree is a multiple of the characteristic of the base field.

Additive case

where all exponents of the polynomial are powers of the characteristic of the base field; this is a special case of the previous one.

We measure the degree of our understanding by our ability to count the number of decomposable polynomials at a given degree. The crucial ingredient for the obvious inclusion-exclusion formula is an understanding of collisions, that is of distinct decompositions of a given polynomial.

Submodules

add_count.sage

for all additive polynomials this gives the exact number of $k$-collisions and by inclusion-exclusion the number of decomposables; see [GGZ10], compare to [CHH04].

add_decompose.sage

for a single additive polynomial, this determines the structure of (the lattice of) right components via the rational Jordan form of the Frobenius on its root space; see [GGZ10] and [F11].

compose_by_brute_force.sage

create look-up tables of complete decompositions (if feasible) or count at least the number of decomposables; output to ./data/

p2_decompose.sage

for the (basic) wild case, create dictionaries to classify decompositions; see [BGZ13].

p2_construct_collision_by_parameter.sage

for the (basic) wild case, create polynomials with a given number of decompositions TODO

tame_collisions.sage

for the tame case, determine the structure (and number) of all tame collisions according to [Z14].

Usage

Load the module in your local Sage installation:

$ sage -q
sage: load('add_count.sage')

See each module's documentation for further instructions.

Todos

• add the formulas of [BGZ13] to p2_construct_collision_by_parameter.sage
• compare with the ore-package and sigma(a) = a^r

Requirements

This code requires the free mathematical software [Sage] which is available for download at http://www.sagemath.org and as cloud service at https://cloud.sagemath.org. It has been tested under GNU/Linux with Sage 6.4.

References

[BGZ13]

(1, 2) Raoul Blankertz, Joachim von zur Gathen & Konstantin Ziegler (2013). Compositions and collisions at degree p2. Journal of Symbolic Computation 59, 113–145. ISSN 0747-7171. URL http://dx.doi.org/10.1016/j.jsc.2013.06.001. Also available at http://arxiv.org/abs/1202.5810. Extended abstract in Proceedings of the 2012 International Symposium on Symbolic and Algebraic Computation ISSAC ’12, Grenoble, France (2012), 91–98.

[CHH04]

Robert S. Coulter, George Havas & Marie Henderson (2004). On decomposition of sub-linearised polynomials. Journal of the Australian Mathematical Society 76(3), 317–328. URL http://dx.doi.org/10.1017/S1446788700009885.

[F11]

Harald Fripertinger (2011). The number of invariant subspaces under a linear operator on finite vector spaces. Advances in Mathematics of Communications 5(2), 407–416. ISSN 1930-5346. URL http://dx.doi.org/10.3934/amc.2011.5.407.

[GGZ10]

(1, 2) Joachim von zur Gathen, Mark Giesbrecht & Konstantin Ziegler (2010). Composition collisions and projective polynomials. Statement of results. In Proceedings of the 2010 International Symposium on Symbolic and Algebraic Computation ISSAC ’10, Munich, Germany, edited by Stephen Watt, 123–130. ACM Press. URL http://dx.doi.org/10.1145/1837934.1837962. Preprint available at http://arxiv.org/abs/1005.1087.

[Sage]

W. A. Stein et al. (2014). Sage Mathematics Software (Version 6.4). The Sage Development Team. URL http://www.sagemath.org.

[Z14]

Konstantin Ziegler (2014). Tame decompositions and collisions. In Proceedings of the 2014 International Symposium on Symbolic and Algebraic Computation ISSAC ’14, Kobe, Japan, edited by Katsusuke Nabeshima, 421–428. ACM Press, Kobe, Japan. URL http://dx.doi.org/10.1145/2608628.2608653. Preprint available at http://arxiv.org/abs/1402.5945.

Author

• Konstantin Ziegler (2013-12-24): initial version

License

This program is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 3 of the License, or (at your option) any later version.

This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

You should have received a copy of the GNU General Public License along with this program. If not, see http://www.gnu.org/licenses/.