Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: Algebraically Solvable Problems: Describing Polynomials as Equivalent to Explicit Solutions. | Algebraically Solvable Problems Describing Polynomials as Equivalent to Explicit Solutions Uwe Schauz Department of Mathematics University of Tbingen Germany Submitted Nov 14 2006 Accepted Dec 28 2007 Published Jan 7 2008 Mathematics Subject Classifications 41A05 13P10 05E99 11C08 11D79 05C15 15A15 Abstract The main result of this paper is a coefficient formula that sharpens and generalizes Alon and Tarsi s Combinatorial Nullstellensatz. On its own it is a result about polynomials providing some information about the polynomial map P xix-xXn when only incomplete information about the polynomial P Xi . Xn is given. In a very general working frame the grid points x 2 X1 X X Xn which do not vanish under an algebraic solution - a certain describing polynomial P Xi . Xn - correspond to the explicit solutions of a problem. As a consequence of the coefficient formula we prove that the existence of an algebraic solution is equivalent to the existence of a nontrivial solution to a problem. By a problem we mean everything that owns both a set S which may be called the set of solutions and a subset Striv u S the set of trivial solutions. We give several examples of how to find algebraic solutions and how to apply our coefficient formula. These examples are mainly from graph theory and combinatorial number theory but we also prove several versions of Chevalley and Warning s Theorem including a generalization of Olson s Theorem as examples and useful corollaries. We obtain a permanent formula by applying our coefficient formula to the matrix polynomial which is a generalization of the graph polynomial. This formula is an integrative generalization and sharpening of 1. Ryser s permanent formula. 2. Alon s Permanent Lemma. 3. Alon and Tarsi s Theorem about orientations and colorings of graphs. Furthermore in combination with the Vigneron-Ellingham-Goddyn property of planar n-regular graphs the formula contains as very special cases 4. Scheim s formula for the .
