= Rational univariate representation
A <rational univariate representation> expresses a finite algebraic solution set using roots of one univariate <polynomial> and rational coordinate <functions>. For distinct solutions, one may use a separating parameter $t$, a square-free <polynomial> $p(t)$, and formulas $x_i=q_i(t)/q_0(t)$ with $q_0$ nonzero at the relevant roots. Recovering multiplicities requires additional care beyond merely listing the distinct points.
For $x^2-2=0$ and $(x-1)y-1=0$, take $p(t)=t^2-2$, $x=t$ and $y=1/(t-1)$. Neither root of $p$ equals one, so both coordinate formulas are defined. Solving one <polynomial> therefore recovers both solutions of the original two-equation system.<Rational Univariate Representation (Lecture 12) by Jan Verschelde>[Section 1, equation (1), separating linear form and finite solution set; example calculated here]{ref}
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