[seqfan] Representing GF(2)[x,y] polynomials as integers

Peter Munn techsubs at pearceneptune.co.uk
Sat Jan 4 11:33:53 CET 2020

I am comparatively new to polynomial rings. I have been looking at them
only because I recognised that an interesting variant of Re'my Sigrist's
A306697 would be a binary operation isomorphic to multiplication in
GF(2)[x,y], and this seemed significant.

OEIS seems to have plenty of sequences, such as A048720, that represent
GF(2)[x] polynomials with integers, and in such a way that binary
exclusive-OR represents polynomial addition. However, I find no precedents
in OEIS for what I propose with GF(2)[x,y] in http://oeis.org/A329329.

Does anyone know of anyone doing anything similar before? I would
especially welcome knowing if anyone has done exactly as I have, except
for assigning x and y the opposite way - as I would swap my use of x and y
to conform, and I'd rather do that sooner than later.

Most likely (I think) someone might have used a pairing function to map x
and y dimensions onto bits of a simple binary encoding. This way, binary
exclusive-OR continues to represent polynomial addition. Such a choice
would complement what I do in A329329, so I have, more tentatively,
started creating a pairing function variant of A329329 (in A329331). The
choice of pairing function seems not clear cut, and if there is an already
known representation based on a different choice, I would change A329331
to conform.

Thanks for any information.


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