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MathGroup Archive 2010

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Working with polynomials in Z/23

  • To: mathgroup at smc.vnet.net
  • Subject: [mg106566] Working with polynomials in Z/23
  • From: me13013 <me13013 at gmail.com>
  • Date: Sat, 16 Jan 2010 06:13:36 -0500 (EST)

I'm exploring polynomials over integer values modulo 23 (i.e. 'x' in
the polynomial can only take integer values, and my polynomials only
have integer coefficients). I know that I can reduce the polynomial
coefficients mod 23 using PolynomialMod.  However, I also want to
reduce the exponents for the identity x^23 = x.  Is there an easy way
to do that?

For example, if I have the polynomial P(x) = x^5 + 2x.  Then
P(P(x)) = x^25 + 10x^21 + 40x^17 + 80x^13 + 80x^9 + 34x^5 + 4x.
PolynomialMod will reduce this to
P(P(x)) = x^25 + 10x^21 + 17x^17 + 11x^13 + 11x^9 + 11x^5 + 4x.
But since x^25 = x^3, the answer I want is
P(P(x)) = 10x^21 + 17x^17 + 11x^13 + 11x^9 + 11x^5 + x^3 + 4x.

I tried to figure out a way to do this using CoefficientList, but I'm
not proficient enough as an occasional Mathematica user to figure out
how to sort of "fold" the list onto itsefl and sum the columns.

Any help would be appreciated,
Bob H


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