---
tag: 681HPCS6G6
kind: definition
name: Unisolvent point set
msc: [41A05, 41A63, 65D05]
defines: 5276MDSDQV
uses: [000AYPA9AJ, 03GSZ8RTZJ, 07GNX8GCYP]
sources:
  - url: https://en.wikipedia.org/wiki/Unisolvent_point_set
---

## Definition

Let $\Omega$ be a set and let $V$ be a [[vector space|000AYPA9AJ]] of functions $\Omega \to \R$ of finite [[dimension|03GSZ8RTZJ]] $N$.
Points $x_1, \dots, x_N \in \Omega$ are *unisolvent* for $V$ if for every $(y_1, \dots, y_N) \in \R^N$ there is exactly one $f \in V$ with $f(x_i) = y_i$ for $i = 1, \dots, N$.

## Remarks

Equivalently, the only $f \in V$ with $f(x_1) = \cdots = f(x_N) = 0$ is $f = 0$.
Equivalently, for a basis $\varphi_1, \dots, \varphi_N$ of $V$, the $N \times N$ matrix $(\varphi_j(x_i))$ is invertible.
The usual case is $V$ equal to the [[polynomials in $d$ variables|07GNX8GCYP]] of total degree at most $n$, where $N = \binom{n + d}{d}$.

Example: in one variable, the points $0, 1, 2$ are unisolvent for the polynomials of degree at most $2$.
A polynomial of degree at most $2$ with three roots is $0$, so the equivalent condition holds.

Non-example: for $V = \set{a + bx + cy : a, b, c \in \R}$ on $\R^2$, the points $(0, 0)$, $(1, 1)$, $(2, 2)$ are not unisolvent.
The nonzero function $x - y \in V$ vanishes at all three points.