Maple command for algebraic combinaiton of polynomial in $\mathbb{Z}_p[x]$

Question

Let $F(x) , f_1(x),\ldots, f_r(x)$ be polynomials with coefficients in $\mathbb{Z}_p$ ($p$ is odd prime)

Is there a Maple command to determine if$$ F(x) = a_1(x)f_1(x)+\dots+ a_r(x)f_r(x) \pmod{p} $$ for some polynomials $a_1(x),\dots,a_r(x)$ with coefficients in $\mathbb{Z}_p$?

Thanks for any help you can give.

Answer 1

Yes. Maple has a polynomial ideal package that implements ideals in polynomial rings over fields. I've never actually used Maple before, but it looks like you can construct what you want by doing something along the lines of the examples here in particular I would guess the following would do it (My example uses three polynomials over $\mathbb Z_3$ to ensure actual machine commands):

with(PolynomialIdeals):
I:=PolynomialIdeal(f1,f2,f3,characteristic=3)
IdealMembership(f,I)

I'm a little confused by Maple's documentation in this instance because it's not clear to me what field it picks when you say characteristic=3. So I'm going to take this moment to demonstrate it in my preferred computer algebra language, Sage:

K=GF(3)
R=PolynomialRing(K,'x')
I=R.ideal(f1,f2,f3)
f in I