The Budan-Fourier theorem gives an upper bound for the number of zeros (with multiplicity) of a polynomial $f(x)$ of degree $n$ in the interval $(a,b]$ in terms of the number of sign variations in the vector of derivatives $$D_{f(x)} = ( f(x), f'(x), f''(x),\ldots, f^{(n)}(x) )$$ at $x=a$ and $x=b$. One proof of the Budan-Fourier theorem considers the multiplicity vector $$M_{f(x)} = ( \mu_0(x), \mu_1(x), \ldots, \mu_n(x) )$$ where $\mu_j(x)$ is the multiplicity of $x$ as a root of the $j$th derivative $f^{(j)}(x)$.
The inverse problem asks: What vectors are the multiplicity vectors of polynomials, and, given a multiplicity vector, what are the associated polynomials? The simultaneous study of multiplicities of real numbers $x_1,\ldots, x_m$ leads to multiplicity matrices and their associated polynomials.
Mel Nathanson