# Unisolvent point set

{{ safesubst:#invoke:Unsubst||$N=Unreferenced |date=__DATE__ |$B= {{#invoke:Message box|ambox}} }} In approximation theory, a finite collection of points ${\displaystyle X\subset R^{n}}$ is often called unisolvent for a space ${\displaystyle W}$ if any element ${\displaystyle w\in W}$ is uniquely determined by its values on ${\displaystyle X}$.
${\displaystyle X}$ is unisolvent for ${\displaystyle \Pi _{n}^{m}}$ (polynomials in n variables of degree at most m) if there exists a unique polynomial in ${\displaystyle \Pi _{n}^{m}}$ of lowest possible degree which interpolates the data ${\displaystyle X}$.
Simple examples in ${\displaystyle R}$ would be the fact that two distinct points determine a line, three points determine a parabola, etc. It is clear that over ${\displaystyle R}$, any collection of k + 1 distinct points will uniquely determine a polynomial of lowest possible degree in ${\displaystyle \Pi ^{k}}$.