How to decide for which values a polynomial system of equations do not have any solutions?

Question

Suppose we have for example a polynomial system of equations $\begin{cases} v_1 &= x_1y_2 - y_1x_2\\ v_2 &= x_1y_3 - y_1x_3\\ v_3 &= x_1y_4 - y_1x_4\\ v_4 &= x_2y_3 - y_2x_3\\ v_5 &= x_2y_4 - y_2x_4\\ v_6 &= x_3y_4 - y_3x_4\\ \end{cases}$

over the field of real numbers $\mathbb{R}$.

How could we decide for which values of $v_1,\dots,v_6$ the system does not have solutions? Could, for example, Gröbner bases be applied here? If so, how?

Answer 1

In general you can try to eliminate the variables which are not parameters. In SageMath:

sage: R.<x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4,v_1,v_2,v_3,v_4,v_5,v_6> = PolynomialRing(QQ)
sage: I = R.ideal(v_1 - (x_1*y_2 - y_1*x_2), v_2 - (x_1*y_3 - y_1*x_3), v_3 - (x_1*y_4 - y_1*x_4), v_4 - (x_2*y_3 - y_2*x_3), v_5 - (x_2*y_4 - y_2*x_4), v_6 - (x_3*y_4 - y_3*x_4))
sage: I.elimination_ideal([x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4]).gens()
[v_3*v_4 - v_2*v_5 + v_1*v_6]

Algorithmically this is done e.g. by computing a Gröbner basis with respect to an elimination ordering in which the $x_j$ and $y_k$ are greater than all $v_i$, and then taking only those generators that contain only $v_i$'s:

sage: R.<x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4,v_1,v_2,v_3,v_4,v_5,v_6> = PolynomialRing(QQ, order='degrevlex(8),degrevlex(6)')
sage: I = R.ideal(v_1 - (x_1*y_2 - y_1*x_2), v_2 - (x_1*y_3 - y_1*x_3), v_3 - (x_1*y_4 - y_1*x_4), v_4 - (x_2*y_3 - y_2*x_3), v_5 - (x_2*y_4 - y_2*x_4), v_6 - (x_3*y_4 - y_3*x_4))
sage: [g for g in I.groebner_basis() if all(v in [v_1,v_2,v_3,v_4,v_5,v_6] for v in g.variables())]
[v_3*v_4 - v_2*v_5 + v_1*v_6]

The equations obtained by setting those generators equal to zero provide necessary conditions upon the parameters for the equations to be solvable.

In your concrete case, the problem of eliminating the variables $x_1,x_2,x_3,x_4,y_1,y_2,y_3,y_4$ is also a problem of implicitization: trying to find equations for the smallest variety that contains the image of the parametrization.

You can read about this stuff e.g. in Chapter 3 Elimination theory of the book Ideals, Varieties, and Algorithms by Cox, Little, and O'Shea.