The Jacobian Conjecture as a Problem of Perturbative Quantum Field Theory

The Jacobian conjecture is an old unsolved problem in mathematics, which has been unsuccessfully attacked from many different angles. We add here another point of view pertaining to the so called formal inverse approach, that of perturbative quantum field theory.Comment: 22 pages, 13 diagram

The Jacobian conjecture: ideal membership questions and recent advances

The Jacobian conjecture can be reduced to the consideration of polynomial maps F:Cn→Cn of the special form F=X−H, where X is the identity map and H contains only higher order terms (i.e., terms of degree at least 2). In that case, it asserts that if |J(F)|, the Jacobian determinant of F, is identically 1, then F is bijective, with a polynomial inverse. The introduction contains a convenient summary of known reductions to yet more special forms and known partial results for those forms. The reductions are formulated as families of conjectures for each pair (n,d), where n≥2 is the dimension and d≥2 is a degree bound. The five cases discussed are: each Hi has degree at most d; each Hi is homogeneous of degree d (or zero); each Hi is of the form Ldi (with Li a linear form) [degree d Drużkowski form]; the Jacobian matrix J(H) is symmetric; and, finally, each Hi is homogeneous of degree d and J(H) is symmetric. In all these five cases, the stated conjectures are equivalent to the Jacobian conjecture if they are true for a fixed d≥3 and all values of n≥2. The generic formal map of special form F=X−H in dimension n has (new) indeterminates as coefficients of the monomials (of all degrees) of H. Its formal inverse has special form, with coefficients that are polynomials in the coefficients of the generic formal map. Previous work has shown that there are explicit formulas for the coefficients of the inverse as sums of monomials indexed by isomorphism classes of certain finite trees. In the five cases above, the generic formal map can be specialized; for instance, for the case of a symmetric J(H), one can write F=X−∇P, where P is a potential function in n variables of degree d+1, and then the inverse coefficients are polynomials in the (indeterminate) coefficients of P. The individual conjectures for a given case and fixed n and d then hold if the inverse coefficients of monomials of sufficiently high degree belong to the radical of an appropriate ideal; the ideals in question are those generated by the coefficients of the monomials of 1−|J(H)| (the Jacobian ideal) or the ideal generated by the coefficients of the monomials of J(H)n (the nilpotency ideal—used in homogeneous cases). Membership and ideals are considered in the Q-algebra generated by the finitely many (because (n,d) is fixed) indeterminates of the case specific generic formal map. Of particular interest are specialized formulas for the formal inverse, and the determination of cases in which the use of the radical can be dropped. The resulting plethora of ideal membership questions are answered in some specific cases by use of a computer algebra system (details not shown). Some explicit case specific formulas are cited or developed for inverse coefficients, homogeneous components of the inverse, and generators for ideals to whose radicals inverse coefficients must belong. Bounds on the degrees of monomials for which inverse coefficient ideal membership must actually be checked are improved. As a sample of results, the last section states that if F=X−H has H homogeneous of degree d, with J(H) symmetric and J(H)3=0, then F is invertible and the degree of its inverse is less than 2d (independent of n)

Random planar trees and the Jacobian conjecture

We develop a probabilistic approach to the celebrated Jacobian conjecture, which states that any Keller map (i.e. any polynomial mapping $F\colon \mathbb{C}^n \to \mathbb{C}^n$ whose Jacobian determinant is a nonzero constant) has a compositional inverse which is also a polynomial. The Jacobian conjecture may be formulated in terms of a problem involving labellings of rooted trees; we give a new probabilistic derivation of this formulation using multi-type branching processes. Thereafter, we develop a simple and novel approach to the Jacobian conjecture in terms of a problem about shuffling subtrees of $d$-Catalan trees, i.e. planar $d$-ary trees. We also show that, if one can construct a certain Markov chain on large $d$-Catalan trees which updates its value by randomly shuffling certain nearby subtrees, and in such a way that the stationary distribution of this chain is uniform, then the Jacobian conjecture is true. Finally, we show that the subtree shuffling conjecture is true in a certain asymptotic sense, and thereafter use our machinery to prove an approximate version of the Jacobian conjecture, stating that inverses of Keller maps have small power series coefficients for their high degree terms.Comment: 36 pages, 4 figures. Section 2.5 added, Section 3 expanded, further minor edit

Faà di Bruno's formula and inversion of power series

Faà di Bruno's formula gives an expression for the derivatives of the composition of two real-valued functions. In this paper we prove a multivariate and synthesised version of Faà di Bruno's formula in higher dimensions, providing a combinatorial expression for the derivatives of chain compositions of functions in terms of sums over labelled trees. We give several applications of this formula, including a new involution formula for the inversion of multivariate power series. We use this framework to outline a combinatorial approach to studying the invertibility of polynomial mappings, giving a purely combinatorial restatement of the Jacobian conjecture. Our methods extend naturally to the non-commutative case, where we prove a free version of Faà di Bruno's formula for multivariate power series in free indeterminates, and use this formula as a tool for obtaining a new inversion formula for free power series

Combinatorial Approaches To The Jacobian Conjecture

The Jacobian Conjecture is a long-standing open problem in algebraic geometry. Though the problem is inherently algebraic, it crops up in fields throughout mathematics including perturbation theory, quantum field theory and combinatorics. This thesis is a unified treatment of the combinatorial approaches toward resolving the conjecture, particularly investigating the work done by Wright and Singer. Along with surveying their contributions, we present new proofs of their theorems and motivate their constructions. We also resolve the Symmetric Cubic Linear case, and present new conjectures whose resolution would prove the Jacobian Conjecture to be true