tailieunhanh - Báo cáo toán học: "On Catalan Trees and the Jacobian Conjecture"

Tuyển tập các báo cáo nghiên cứu khoa học hay nhất của tạp chí toán học quốc tế đề tài: On Catalan Trees and the Jacobian Conjecture. | On Catalan Trees and the Jacobian Conjecture Dan Singer Oakland University Rochester MI dwsinger@ Submitted July 11 2000 Accepted November 28 2000 Abstract New combinatorial properties of Catalan trees are established and used to prove a number of algebraic results related to the Jacobian conjecture. Let F x1 H1 x2 H2 . x Hn be a system of n polynomials in C x1 x2 . xn the ring of polynomials in the variables x1 x2 . x over the field of complex numbers. Let H H1 H2 . Hn . Our principal algebraic result is that if the Jacobian of F is equal to 1 the polynomials Hi are each homogeneous of total degree 2 and Ặ 3 0 then HoHoH 0 and F has an inverse of the form G G1 G2 . Gn where each Gi is a polynomial of total degree 6. We prove this by showing that the sum of weights of Catalan trees over certain equivalence classes is equal to zero. We also show that if all of the polynomials Hi are homogeneous of the same total degree d 2 and Xi 2 0 then H o H 0 and the inverse of F is G x1 H1 . x Hn . 1 Introduction Let Fl F2 . Fn be polynomials in C x1 x2 . xn the ring of polynomials in the variables X1 X2 . xn over the field of complex numbers. The Jacobian conjecture states that if the Jacobian of the system F Fl F2 . Fn is equal to a non-zero scalar number then there exists an inverse system of polynomials G G1 G2 . G n such that Gi F1 F2 . . Fn Xị for each i n. For example let n 2 and consider Fl X1 xi X2 2 F2 X2 - xi X2 2. Keywords Catalan trees Jacobian conjecture formal tree expansions AMS Subject Classifications 05E99 primary 05A99 05C05 14R15 secondary THE ELECTRONIC JOURNAL OF COMBINATORICS 8 2001 R2 1 Since F1 - F1 F2 2 X1 and F2 Fi F2 2 X2 the inverse to the system F Fl F2 is the system G G1 G2 defined by G1 X1 X1 X2 2 G2 X2 X1 X2 2. Note that the Jacobian of F is - 6F1 dx @F1 - @x r 1 2x1 2x2 2x1 2x2 det det ôF2 - @X1 @F2 dx2 - 2x1 2x2 1 2X1 2X2 1. There are a number of partial results relating to systems in which Fi xi Hi for all i where each Hi is .