Reduction of the Multivariate Quadratic (MQ) System to a Trivial Equation

This shows a method to reduce an NP-complete problem to a trivial equation. Since it does not use a Grover basis and is polynomial—because it relies on FullSimplify and Resultant—it could be considered that we have achieved a practical P=NP.

Used Formulas for Wolfram Alpha

FullSimplify[ {a11*xPower[1,2] + a12*x1*x2 + a22*xPower[2,2] + b1*x1 + b2*x2 + c1, d11*xPower[1,2] + d12*x1*x2 + d22*xPower[2,2] + e1*x1 + e2*x2 + c2} ]

FullSimplify[ {a11*xPower[1,2] + a12*x1*x2 + a22*xPower[2,2] + b1*x1 + b2*x2 + c1, d11*xPower[1,2] + d12*x1*x2 + d22*xPower[2,2] + e1*x1 + e2*x2 + c2}{a1*Power[x,2] + a2*x*y + a3*Power[y,2] + b1*x + b2*y + c1, d1*Power[x,2] + d2*x*y + d3*Power[y,2] + e1*x + e2*y + c2}] ]

FullSimplify[{a1*Power[x,2] + a2*x*y + a3*Power[y,2] + b1*x + b2*y + c1, d1*Power[x,2] + d2*x*y + d3*Power[y,2] + e1*x + e2*y + c2}]

Resultant[a1*Power[x,2] + a2*x*y + a3*Power[y,2] + b1*x + b2*y + c1, d1*Power[x,2] + d2*x*y + d3*Power[y,2] + e1*x + e2*y + c2, y]