# Iterate over multivariate polynomials over finite fields

Say we have a finite field, e.g. $F_4$, and consider the $n$-ary polynomials $R=F_4[x_1,\dots,x_n]$ over this field. I want to iterate over all these polynomials in $R$. Since the polynomials are over a finite field there are only finitely many different polynomials (considered as functions $F_4^n \to F_4$). How can I do this? For $n=1$ I could do

R.<x> = PolynomialRing(GF(4))
S.<a> = R.quo(sage.rings.ideal.FieldIdeal(R))
S.is_finite()


Then I could iterate over $S$ and simply lift all the elements from $S$ back to $R$, i.e. s.lift(). The same thing however does not work for several polynomials:

R.<x,y> = PolynomialRing(GF(4))
S.<a,b> = R.quo(sage.rings.ideal.FieldIdeal(R))
S.is_finite()


yields the error

AttributeError: 'super' object has no attribute 'is_finite'

As an alternative I could manually generate all multivariate polynomials with exponents less than the order of the field. However, this seems quite tedious and like a very "un-sage"/not algebraic way.

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A vector space basis for $R/I$ is also called a normal basis for $I$. You can obtain it as follows:

sage: R.<x,y> = PolynomialRing(GF(4))
sage: NB = sage.rings.ideal.FieldIdeal(R).normal_basis(); NB
[x^3*y^3, x^2*y^3, x*y^3, y^3, x^3*y^2, x^2*y^2, x*y^2, y^2, x^3*y, x^2*y, x*y, y, x^3, x^2, x, 1]


Then you want to give these monomials all possible coefficients:

sage: from itertools import product
sage: [sum(c*m for (c,m) in zip(C,NB)) for C in product(R.base_ring(), repeat=len(NB))]


The output consists of $4^{16} = 4294967296$ polynomials.

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