Whe you write :
sage: var('a0,a1') ; R.<x,y> = QQ[] ;
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x ;
The product a1*x (and the other elementary operations) is done using coercion, that is Sage first searches the common parent between the symbolic ring and the bivariate rational polynomials in x,y. It turns out that it is the symbolic ring itself. Hence, what you get is and element of the symbolic ring:
sage: Pol.parent()
Symbolic Ring
In particular, you lose the polynomial structure in x,y.
The first approach is to let Sage consider a0 and a1 a possible coefficients for polynomials in x,y, by declaring the polynomial ring R to be defined over the symbolic ring SR:
sage: var('a0,a1') ; R.<x,y> = SR[]
(a0, a1)
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Symbolic Ring
Then you can extract the nonzero coefficients:
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: solve(Pol.coefficients(),[a0,a1])
[]
There is of course no solution since a0 should be both equal to 0 and -5.
Another approach is to let the coefficients be elements of another polynomial ring S, that will be the ring over which R will be defined:
sage: S.<a0,a1> = QQ[] ; S
Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: R.<x,y> = S[] ; R
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: Pol = (a1*x+a0)*(1-x*y)-5*y*x
sage: Pol
(-a1)*x^2*y + (-a0 - 5)*x*y + a1*x + a0
sage: Pol.parent()
Multivariate Polynomial Ring in x, y over Multivariate Polynomial Ring in a0, a1 over Rational Field
Now, the common zeroes of the coeficients of Pol (which are polynomials, elements of S), is nothing but the variety of an ideal:
sage: Pol.coefficients()
[-a1, -a0 - 5, a1, a0]
sage: S.ideal(Pol.coefficients())
Ideal (-a1, -a0 - 5, a1, a0) of Multivariate Polynomial Ring in a0, a1 over Rational Field
sage: I = S.ideal(Pol.coefficients())
sage: I.variety()
[]
Which is empty as well.
