I am studying iterated preimage structures of functions on a finite set.

The main structure of interest to me, the Sigma Matrix, is derived from a matrix listing the element-wise preimage sets at increasing inverse depth. This intermediate matrix is the "preimage matrix" (shown as $P$ below)

$P=\left(\begin{array}{cccc} f^{-1}(x_{1}) & f^{-2}(x_{1}) & \cdots & f^{-n}(x_{1})\\ f^{-1}(x_{2}) & \ddots\\ \vdots\\ f^{-1}(x_{n}) & & & f^{-n}(x_{n}) \end{array}\right)$

We then look at the sizes of such matrix elements. Yielding the "sigma" matrix $\Sigma$ with entries

$\Sigma=\left(\begin{array}{cccc} \mid f^{-1}(x_{1})\mid & \cdots & \mid f^{-n}(x_{1})\mid \\ &\ddots\\ \vdots\\ \mid f^{-1}(x_{n})\mid & & \mid f^{-n}(x_{n})\mid \end{array}\right)$

  • Result 1: Each column forms an integer partition of n = dom(f) since each column of the sigma matrix has sum n.

Let each column of $\Sigma$ be represented by its own polynomial where: if $col(1) = [a,b,c]$ then the associated polynomial for $\Sigma_{X1}$ is
$y_{1}=ax^{0}+bx^{1} + cx^{2}$

  • Result 2: Now taking all the columns,for any size sigma matrix (on any size domain) and deriving all the polynomials, they have a solution at $(1,n)$.

Question: There seems to always be one and only one solution to these systems of equations. Id appreciate a reference to this sort of exploration

For additional background please see my last question on the sigma matrices: previous question


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