zbMATH — the first resource for mathematics

A polynomial approach to cocycles over elementary Abelian groups. (English) Zbl 1193.20064
Any map $$\text{GF}(q)\times\text{GF}(q)\to\text{GF}(q)$$ may be represented uniquely as a polynomial in $$\text{GF}(q)[x,y]$$ of $$x$$-degree and $$y$$-degree at most $$q-1$$; here $$q=p^n$$, $$p$$ a prime. The authors apply this to maps $$\mathbb{Z}_p^n\times\mathbb{Z}_p^n\to\mathbb{Z}_p^n$$. They determine which such polynomials correspond to cocycles and which correspond to coboundaries. When $$p=2$$, the case of most applications, bases are found for cocyles and coboundaries. A consequence is that each cocycle has a unique representation as a sum of a coboundary and a multiplicative cocycle of a certain type.

MSC:
 20J06 Cohomology of groups 11T06 Polynomials over finite fields
Full Text:
References:
 [1] DOI: 10.1007/s10623-008-9172-z · Zbl 1178.94190 · doi:10.1007/s10623-008-9172-z [2] DOI: 10.1109/18.850692 · Zbl 0994.94032 · doi:10.1109/18.850692 [3] DOI: 10.1080/00927870008826937 · Zbl 0999.20047 · doi:10.1080/00927870008826937 [4] DOI: 10.2307/2304500 · Zbl 0030.11102 · doi:10.2307/2304500 [5] DOI: 10.1023/A:1008292303803 · Zbl 0872.51007 · doi:10.1023/A:1008292303803 [6] DOI: 10.2307/2323743 · Zbl 0553.05004 · doi:10.2307/2323743 [7] DOI: 10.1007/3-540-45311-3_1 · doi:10.1007/3-540-45311-3_1 [8] Lucas, Théorie des Nombres, Tome premier pp 417– (1891) [9] DOI: 10.1109/TIT.2005.864481 · Zbl 1177.94136 · doi:10.1109/TIT.2005.864481 [10] Lidl, Finite Fields (1997) [11] Brown, Cohomology of Groups (1982) · doi:10.1007/978-1-4684-9327-6 [12] Batten, J. Aust. Math. Soc. 82 pp 297– (2007) [13] DOI: 10.1016/j.ffa.2006.03.003 · Zbl 1170.94009 · doi:10.1016/j.ffa.2006.03.003 [14] Horadam, Appl. Algebra Engrg. Comm. Comput. 14 pp 65– (2003) [15] DOI: 10.1023/A:1019999223151 · Zbl 1027.05013 · doi:10.1023/A:1019999223151 [16] Horadam, Hadamard Matrices and Their Applications (2006)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.