Session S14 - Global Injectivity, Jacobian Conjecture, and Related Topics

Monday, July 12, 12:00 ~ 12:50 UTC-3

The Jacobian conjecture through the lens of formal inverse and combinatorics

David Wright

Washington University in St. Louis, USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

One approach to the celebrated Jacobian conjecture is to consider the formal power series of a map satisfying the Jacobian hypothesis and to try to show that this system of power series is actually a polynomial map, i.e., all but finitely many of its summands are zero. It generally begins by taking a map of cubic homogeneous type and studying the homogeneous summands of its formal inverse and the polynomial conditions that say the jacobian determinant is 1 (``Jacobian constraints"). The conjecture thereby becomes equivalent to a statement about ideal membership in a polynomial ring. Combinatorics enters the scene in understanding the terms of the formal inverse and the Jacobian constraints. This line of attack has yielded a number of partial results over the years. We will explain why the approach is tantalizingly compelling, but also has glaring weaknesses that leave us trying to prove, using formal inverse and combinatorics, known results that have been proved by other methods.

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