Is it possible to determine the coefficients of two polynomials, if we are given 2n different points at which they cross each other ?
In other words, If $f(p) = \sum_{i}\alpha_{i}p^{i}$ and $g(p) = \sum_{i}\beta_{i}p^{i}$ are two $n-$degree polynomials and we have $\{p_{1}, p_{2}, \dots, p_{2n}\}$ such that:
$v_{1}f(p_{1}) = g(p_{1})$
$v_{2}f(p_{2}) = g(p_{2})$
$\vdots$
$v_{2n}f(p_{2n}) = g(p_{2n})$
where $v_{1}, \dots, v_{2n}$ are all known constants, $v_{i}\neq v_{j}$.
Is that enough to determine $\{\alpha_{0}, \dots, \alpha_{n}, \beta_{0}, \dots, \beta_{n}\}$ ?
I understand each crossing would provide us a linear constraint on the coefficients. Hence my intuition would be that 2n different crossings give 2n linearly independent constraints. But then, when I try examples or even try to express all the constraints as a matrix form, I get a homogeneous system of linear equations, whose only solution turns out to be $\alpha_{i} = 0 = \beta_{i}$.
Can anyone please help.