I have an ideal $I$ generated by multivariate polynomials $f,g$ over BooleanPolynomialRing. Suppose another polynomial $h$ is in $I$. So there are two polynomials $h_1$ and $h_2$ such that $h=h_1 f +h_2 g$. How to find $h_1, h_2$?
h.lift(I) should do the job. See documentation for details.
Since lift is not implemented for BooleanPolynomialRing, we can workaround by considering the ideal generated by $f,g$ and $x_i^2+x_i$ in $GF(2)[x_1,\dots,x_n]$. Here is a sample code:
B.<x,y> = BooleanPolynomialRing()
I = ideal(x+1,y+1)
pol = x*y + 1
#print( pol.lift(I) ) # this is not implemented
#workaround:
F = GF(2)[B.gens()]
I2 = ideal( [F(g) for g in I.gens()] + [g^2+g for g in F.gens()] )
coefs = [B(c) for c in F(pol).lift(I2)[:len(I.gens())]]
print( coefs )
This is not implemented for multivariate polynomials over BooleanPolynomialRing.
Then consider the ideal generated by $f,g$ and $x_i^2+x_i$ of $GF(2)[x_1,\dots,z_n]$ instead. I've added a sample code.
Asked: 2021-05-19 20:10:35 +0100
Seen: 231 times
Last updated: May 20 '21
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