Let $(x_1,y_1),...,(x_n,y_n)\in \mathbb{R}^2 $, where $x_i\neq x_j$ if $i\neq j$. Let $p$ be a polynomial such that $$\det\begin{pmatrix} p(x)& 1 & x & x^2 &\dots & x^n \\ y_0 & 1 & x_0 & x_0^2 &\dots &x_0^n \\ y_1 & 1 & x_1 & x_1^2 &\dots &x_1^n \\ \vdots & & & & & \vdots \\ y_n & 1 & x_n & x_n^2 &\dots & x_n^n \end{pmatrix}=0. $$ Then $p(x_k)=y_k ,\forall k=1,...,n$.

My ideas: We should compute the determinant using Laplace's formula, although I can't see a nice pattern to do a proof by induction or to conclude the proposition.

Thanks for the help




Hint. Put $x=x_0$ and subtract the second row from the first row. The determinant is then equal to $$ \det\pmatrix{p(x_0)-y_0&0_{1\times(n+1)}\\ \ast&V}=(p(x_0)-y_0)\det(V), $$ where $V$ is a Vandermonde matrix. In order that this is zero, ...


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.