A problem concerning the perturbation of roots of a system of homogeneous algebraic equations is investigated. The question of conservation and decomposition of a multiple root into simple roots are discussed. The main theorem on the conservation of the number of roots of a deformed (not necessarily homogeneous) algebraic system is proved by making use of a homotopy connecting initial roots of ...

We introduce a new Steiner-type problem for directed graphs named q-Root Steiner Tree. Here one is given a directed graph G = (V,A) and two subsets of its vertices, R of size q and T , and the task is to find a minimum size subgraph of G that contains a path from each vertex of R to each vertex of T . The special case of this problem with q = 1 is the well known Directed Steiner Tree problem, w...

In this paper, for any positive integer N we shall study the special values of multiple polylogarithms at Nth roots of unity, called multiple polylogarithmic values (MPVs) of level N . By standard conjectures linear relations exist only between MPVs of the same weight. LetMPVZ(w,N) be the Z-module spanned by MPVs of weight w and level N . Our main interest is to investigate for what w and N the...

A given polynomial, possibly with multiple roots, is factored into several lower-degree distinct-root polynomials with natural-order-integer powers. All the roots, including multiplicities, of the original polynomial may be obtained by solving these lowerdegree distinct-root polynomials, instead of the original high-degree multiple-root polynomial directly. The approach requires polynomial Grea...

For any positive integer N let μN be the group of the N th roots of unity. In this note we shall study the Q-linear relations among the values of multiple polylogarithms evaluated at μN . We show that the standard relations considered by Racinet do not provide all the possible relations in the following cases: (i) level N = 4, weight w = 3 or 4, and (ii) w = 2, 7 <N < 50, and N is a power of 2 ...