Logarithms of iteration matrices, and proof of a conjecture by Shadrin and Zvonkine

A proof for a conjecture by Shadrin and Zvonkine, relating the entries of a matrix arising in the study of Hurwitz numbers to a certain sequence of rational numbers, is given. The main tools used are iteration matrices of formal power series and their (matrix) logarithms.Comment: 29 p

Differentially 4-uniform functions

We give a geometric characterization of vectorial boolean functions with differential uniformity less or equal to 4

Analogue of Newton-Puiseux series for non-holonomic D-modules and factoring

We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i. e. non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors