Charles Ching-An Cheng

Younger mates and the Jacobian conjecture

Let F, G E C[x, y]. If the Jacobian determinant of F and G is 1, then G is said to be a Jacobian mate of F. If, in addition, G has degree less than that of F, then G is said to be a younger mate of F . In this paper, a necessary and sufficient condition is given for a polynomial to have a younger mate. This also gives rise to a formula for the younger mate if it exists. Furthermore, a conjecture concerning the existence of a younger mate is shown to be equivalent to the Jacobian conjecture. Throughout this paper, F and G will be polynomials in C[x, y] where C denotes the field of complex numbers. We say that F and G satisfy the Jacobian hypothesis if their Jacobian determinant is one, i.e., Fx Gy Fy Gx = 1 . In this case, we also say that G is a Jacobian mate of F. Furthermore, if the x-degree (resp. y-degree, total degree) of G is less than that of F, then G is said to be a younger mate of F relative to the x-degree (resp. y-degree, total degree). For instance, x + y has younger mates y and -x relative to the x-degree and the y-degree, respectively, but has no younger mate relative to the total degree. This paper was motivated by the Jacobian conjecture which asserts that if F has a Jacobian mate G, then (F, G) is an automorphism pair. In Section 1, it is shown that a younger mate is unique (up to an additive constant) and universal, i.e., if a Jacobian mate G of F exists, then any other mate of F can be expressed as G plus a polynomial in F. In Section 2, the problem of existence of a younger mate of F is reduced to the case where F is monic in both variables. In Section 3, a necessary and sufficient condition for the existence of a younger mate and a formula for a younger mate provided one exists are given. Finally, in Section 4, a conjecture concerning the existence of younger mates is formulated and shown to be equivalent to the Jacobian conjecture. Received by the editors July 6, 1993 and, in revised form, January 18, 1994. 1991 Mathematics Subject Classification. Primary 13B25, 13F20, 14E09, 16W20.

Reversion of power series and the extended Raney coefficients

In direct as well as diagonal reversion of a system of power series, the reversion coefficients may be expressed as polynomials in the coefficients of the original power series. These polynomials have coefficients which are natural numbers (Raney coefficients). We provide a combinatorial interpretation for Raney coefficients. Specifically, each such coefficient counts a certain collection of ordered colored trees. We also provide a simple determinantal formula for Raney coefficients which involves multinomial coefficients. Let F1, . . . , Fn be polynomials in variables x1, . . . , xn with complex coefficients, where n > 2. Suppose, for each i, Fi = xi + higher degree terms and the Jacobian determinant of F1, . . . , Fn is equal to 1. Then the Jacobian Conjecture [1], [9] asserts, in this case, that x1, . . . , xn are also polynomials in F1, . . . , Fn with complex coefficients. This long-standing conjecture has not been solved even for n = 2. Since it can be proved that x1, . . . , xn are (formal) power series in F1, . . . , Fn with complex coefficients, the Jacobian Conjecture asserts that these power series are really polynomials. This provides the motivation for this paper. Let F1, . . . , Fn be power series in variables x1, . . . , xn of the form Fi = xi + higher degree terms with indeterminate coefficients for each i. It is known (e.g. [2, Chapter III, Section 4.4, Proposition 5, p. 219]) that F = (F1, . . . , Fn) has a (unique) compositional inverse, i.e., there exists G = (G1, . . . , Gn) where each Gi is a power series in variables x1, . . . , xn such that F ◦ G = 1 and G ◦ F = 1, or equivalently, Fi(G1, . . . , Gn) = xi and Gi(F1, . . . , Fn) = xi for all i. There are various methods in the literature to find the coefficients of Gi. In this paper we shall present two new ones. Since each coefficient of Gi is a polynomial in the indeterminate coefficients of F1, . . . , Fn, it is enough to find the coefficients of these polynomials. We will refer to these coefficients as the (extended) Raney coefficients. In the first method, generating functions in infinitely many variables are used to show that each Raney coefficient has a combinatorial interpretation as the number of colored trees in a certain collection (Theorems 2.4 and 2.5). In Received by the editors April 4, 1994. 1991 Mathematics Subject Classification. Primary 13F25, 05A15, 05C05, 13P99; Secondary 32A05.

Numerical solutions to the problem of thermoelastic contact of two rods

We present numerical simulations of a model for quasistatic contact of two rods. The problem consists of determining the temperature and displacement fields in two collinear rods each held fixed at one end and free to come in contact at the other. The assumption of a quasistatic process leads to a decoupling of the displacement from the temperature. The problem can be reduced to considering a coupled system of two nonlinear parabolic equations for the temperature, with nonlocal terms. The thermal interaction between the contacting ends is modeled by a coefficient of heat exchange that depends on the gap between the rods when there is no contact and on the contact stress when there is contact. Using the Crank-Nicolson scheme and iterations we show that the model is capable of rather interesting behavior. By appropriate choices of the initial conditions, we obtain solutions with periods of contact and loss of contact. When the boundary temperature is periodic the solution settles very quickly into a periodic pattern of contact and separation. The stability of steady-states is investigated in the case when there are three steady-state solutions. Finally, we compare the solutions of this model with those of the uncoupled model, that is usually employed in applications.

A case of the Jacobian conjecture

A brief and elementary proof is given for a theorem of Bass, Connell and Wright. Suppose F = X + H is a polynomial map. If H is homogeneous of degree ≥2 and has Jacobian matrix whose square is zero, then F is invertible with inverse G = X − H. We also point out an error in a previous published proof of this result.

Quadratic linear Keller maps

Abstract It is proved that a quadratic linear Keller map C n → C n is linearly triangularizable if its rank is at most 2 or corank at most 1.

Endomorphisms of the plane sending linear coordinates to coordinates

Let k be a field of characteristic zero. We show that an endomorphism of k[X1,X2] which sends each linear coordinate to a coordinate is an automorphism of k[X1,X2]. In [3] the authors raised the following question (referred to as Problem 1): let k be a field of characteristic zero and A := k[X1, · · · , Xn] the polynomial ring over k. Is every k-endomorphism of A which sends each coordinate of A to a coordinate of A an automorphism of A? (A polynomial f of A is called a coordinate of A if there exist F2, · · · , Fn in A such that A = k[f, F2, · · · , Fn].) Problem 1 was answered affirmatively for n = 2 in [3]. The case n ≥ 3 remains open for arbitrary k. However, as was observed by Derksen, a negative answer to Problem 1 for algebraically closed fields would give a counterexample to the Jacobian Conjecture. More explicitly, he shows that ([3, Lemma 2.4]) if a k-endomorphism φ of A sends linear coordinates to coordinates, then detJφ(x) 6= 0 for all x ∈ k, so det Jφ ∈ k∗ if k is algebraically closed! (A polynomial in A is called a linear coordinate if it is of the form c1X1 + · · · + cnXn for some ci in k, not all zero.) Based on Derksen’s Lemma, Jelonek[7] gives a positive answer in any dimension to Problem 1 for algebraically closed fields! (He shows that the coordinate preservation property implies that φ is proper which together with detJφ ∈ k∗ shows that φ is an automorphism of A.) In [10], Mikhalev, Yu and Zolotykh, motivated by Derksen’s Lemma, consider the following stronger version of Problem 1, referred to as Problem 2: is every endomorphism of A which sends every linear coordinate to a coordinate an automorphism? Of course (as observed above) in case that k is an algebraically closed field the hypothesis implies that detJφ ∈ k∗. So if the Jacobian Conjecture is true, the answer to Problem 2 is yes. Consequently, a negative solution would give a counter-example to the Jacobian Conjecture. For n = 2 the algebraically closed case was solved affirmatively in [3]. The case n ≥ 3 remained open (if k = k̄). On the other hand, in [10] the authors show that the answer to Problem 2 is negative in case n ≥ 3 and k is any non-algebraically closed field. However, the case n = 2 remained open for k a non-algebraically closed field. Received by the editors December 12, 1997 and, in revised form, September 1, 1998. 1991 Mathematics Subject Classification. Primary 13B25, 13F20, 14E09, 16W20.

Cubic linear Keller maps

Abstract Let F=(X 1 +A 1 3 ,X 2 +A 2 3 ,…,X n +A n 3 ) : C n → C n be a polynomial map with Jacobian determinant 1 where Ai is a linear homogeneous polynomial in X1,X2,…,Xn. Let M denote the coefficient matrix of the linear forms A1,A2,…,An. If rank M≤2 or rank M≥n−2 then F is linearly triangularizable.

On new types of rational rotation-minimizing frame space curves

The existence of rational rotation-minimizing frames (RRMF) on polynomial space curves is characterized by the satisfaction of a certain identity among rational functions. In this note we prove that previously thought degree limitations on that condition are incorrect. In that regard, new types of RRMF curves are discovered.

Power linear Keller maps of dimension three

In this paper it is proved that a power linear Keller map of dimension three over a field of characteristic zero is linearly triangularizable. Let K be a field. A polynomial map F in dimension n over K is an n-tuple (F1, F2, · · · , Fn) of polynomials in K[X1, X2, · · · , Xn]. If G is another polynomial map of the same dimension, then the composition of F and G is defined by F ◦G = (F1(G1, G2, · · · , Gn), · · · , Fn(G1, G2, · · · , Gn)). The polynomial map F is invertible if there exists a polynomial map G such that F ◦G and G◦F are both identities. It is Keller if the determinant of its Jacobian is a nonzero element in K, i.e., det JF ∈ K∗. By the chain rule for Jacobians, invertible polynomial maps are Keller maps. The famous Jacobian conjecture states that if charK = 0, then any Keller map is invertible (see, e.g., [1] or [4]). A polynomial map F is power linear if it is of the form (X1 +A1 1 , X2 +A d2 2 , · · · , Xn + An n ) where Ai is a linear form in X1, X2, · · · , Xn and di ≥ 2 for all i. It is cubic linear if it is power linear where di = 3 for all i. Druzkowski [2] showed that in the case charK = 0 if cubic linear Keller maps are invertible, then the Jacobian conjecture would be true. A polynomial map is triangular if it is of the form (X1 +p1, X2 +p2, · · · , Xn+pn) where pi is a polynomial in K[Xi+1, Xi+2, · · · , Xn]. It is linearly triangularizable if there exists a linear invertible polynomial map φ such that φ ◦ F ◦ φ−1 is triangular. A polynomial map is elementary if it is of the form (X1, · · · , Xi−1, Xi + p,Xi+1, · · · , Xn) where p ∈ K[X1, · · · , Xi−1, X̂i, Xi+1, · · · , Xn]. It is tame if it can be written as a composition of invertible linear maps and elementary maps. It is not hard to see that linearly triangularizable maps are tame and tame maps are invertible. The tame generators conjecture asserts that an invertible polynomial map is tame. This is proved in dimension two by Jung [5] and van der Kulk [6]. For dimensions beyond two, it is an open problem. In this note we prove that power linear Keller maps in dimension three over a field of characteristic zero are linearly triangularizable (hence tame and invertible) proving both the Jacobian and the tame generators conjecture in this case. (It is worth noting that if the degrees of the components of these maps are all equal to three, then the result is a special case of that of Wright [7] which states that all cubic homogeneous polynomial maps are linearly triangularizable.) Received by the editors August 1, 1999 and, in revised form, February 2, 2000. 2000 Mathematics Subject Classification. Primary 14R15, 14R10.

A chain rule for subresultants

Abstract Let I be an integral domain and let f , g , h be polynomials in x over I with positive degrees m , n and k . McKay and Wang proved that res( f ∘ h , g ∘ h )=[lc( h ) mn res( f , g )] k where res( f , g ) denotes the resultant of f and g , and lc( h ) denotes the leading coefficient of h . In this paper we extend the result to that of subresultants.