Shapiro Conjecture 1: The Wronski Map

The Wronski Map

Let f₁(t), f₂(t), ..., f_k(t) be univariate polynomials of degree d.
Their Wronskian is the determinant of the matrix of their derivatives W := det ( f_j^(i-1)(t) ) _{i,j=1,...,k .} This typically has degree k(d+1-k).

Up to a scalar factor, this Wronskian depends only on the linear span of the polynomials
f₁(t), f₂(t), ..., f_k(t). Removing these ambiguities gives the Wronski map

W : Gr(k,d+1) ----> P^k(d+1-k) ,
where Gr(k,d+1) is the Grassmannian of k-dimensional subspaces of polynomials of degree d, and P^k(d+1-k) is the projective space of polynomials of degree k(d+1-k).

Since the Grassmannian has dimension k(d+1-k), we should expect this map to be finite-to-one.