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A good way of thinking about a system of polynomial equations is that the common zero set of polynomials defines an algebraic variety. You want to know its dimension (which might be -1 for the empty set).
sage: R.<x,y> = QQ[]
sage: R.ideal([x*y]).dimension() # solution is x=0 and y=0 plane intersecting at origin
1
sage: R.ideal([x*y,y-1]).dimension() # only solution x=0, y=1
0
sage: R.ideal([x*y,y-1,x-1]).dimension() # inconsistent
-1
