Reduction Theorems for the Strong Real Jacobian Conjecture
RREDUCTION THEOREMS FOR THE STRONG REALJACOBIAN CONJECTURE
L. ANDREW CAMPBELL
Abstract.
Implementations of known reductions of the Strong Real JacobianConjecture (SRJC), to the case of an identity map plus cubic homogeneous orcubic linear terms, and to the case of gradient maps, are shown to preservesignificant algebraic and geometric properties of the maps involved. Thatpermits the separate formulation and reduction, though not so far the solution,of the SRJC for classes of nonsingular polynomial endomorphisms of real n-space that exclude the Pinchuk counterexamples to the SRJC, for instancethose that induce rational function field extensions of a given fixed odd degree. Introduction
The Jacobian Conjecture (JC) [1, 11] asserts that a polynomial map F : k n → k n ,where k is a field of characteristic zero, has a polynomial inverse if it is a Kellermap [16], which means that its Jacobian determinant, j ( F ), is a nonzero element of k . The JC is still not settled for any n > k of characteristiczero. It is well known that it would suffice to prove the JC for k = R and allpositive n . There are many generalizations to endomorphisms of R n [20, 18]. Themost natural is the Strong Real Jacobian Conjecture (SRJC), which asserts thata polynomial map F : R n → R n , has a real analytic inverse if it is nonsingular,meaning that j ( F ), whether constant or not, vanishes nowhere on R n . However,Sergey Pinchuk exhibited a family of counterexamples for n = 2 [19]. They arealso counterexamples to the Rational Real Jacobian Conjecture (RRJC) [7], whichis the extension of the SRJC to include everywhere defined rational nonsingularendomorphisms. Everywhere defined means that each component of the map can beexpressed as the quotient of two polynomials with a nowhere vanishing denominator.Any such F : R n → R n has finite fibers of size at most the degree of the associatedfinite algebraic extension of rational function fields.Let dex denote that extension degree, mfs the maximum fiber size, and sagthe size of the automorphism group of the extension. While dex and mfs are notgenerally equal, they are always of the same parity. The conditions dex odd, mfsodd, and sag = 1 are all necessary for invertibility, and mfs = 1 or dex = 1 issufficient. All the Pinchuk counterexamples satisfy dex = 6, mfs = 2, and sag = 1[5, 6]. Thus the simplest unproved and unrefuted challenge conjecture in this arenais that dex = 3 and sag = 1 is sufficient in the polynomial case.Two known reduction procedures for the SRJC, to the case of maps of cubichomogeneous, or even better cubic linear, type [15, 1, 8, 11] and to the case of a Mathematics Subject Classification.
Primary 14R15; Secondary 14E05 14P10.
Key words and phrases. real rational map, real Jacobian conjecture. a r X i v : . [ m a t h . AG ] J a n L. ANDREW CAMPBELL symmetric Jacobian matrix [17], are shown here to preserve the numerical attributesdex, mfs, and sag.In consequence, the two reductions can be applied to conjectures involving thoseattributes, such as the above challenge conjecture.2.
Background
There are two classic reductions of the ordinary JC to Yagzhev maps [15, 1] andto Dru˙zkowski maps [8]. A Yagzhev map is a polynomial map of the form F = X + H , where X = ( x , . . . , x n ), and each component of H is a cubic homogeneouspolynomial in the variables x , . . . , x n . Yagzhev maps are also called maps of cubichomogeneous type. A Dru˙zkowski map (or map of cubic linear type) is a Yagzhevmap, for which the components of H are cubes of linear forms ( h i = l i ). Ina departure from the convention in some other works, these definitions impose norestriction on j ( F ), beyond the obvious j ( F )(0) = 1. Note, however, that a Yagzhevmap F = X + H is a Keller map if, and only if, H has a Jacobian matrix, J ( H ),that is nilpotent, since both assertions are just different ways of saying that theformal power series matrix inverse of J ( F ) is polynomial.Reduction theorem proofs use the strategy of transforming an original map intoa map of the desired form in a succession of steps that preserve the truth value ofcertain key properties (and typically increase the number of variables).For the JC, C is usually selected as the ground field and the key properties arethe Keller property and the existence of a polynomial inverse. Such proofs thenapply over any ground field of characteristic zero, including R . But the strategyand specific steps can be applied more generally than just to polynomial Kellermaps and yields, for instance, a reduction of the SRJC to the cubic linear case [8].Dru˙zkowski noted this explicitly, with the preserved properties being a nowherevanishing Jacobian determinant and bijectivity.Historical Note. At the 1997 conference in Lincoln, Nebraska, to honor themathematical work of Gary H. Meisters, it was suggested by T. Parthasarathy thatthe SRJC reduction be attempted for the 1994 counterexample of Pinchuk. Thechallenge was taken up by Engelbert Hubbers, and in 1999 he demonstrated theexistence of a counterexample to the SRJC of cubic linear type, coincidentally indimension 1999. He started with exactly the specific Pinchuk map of total degree25 circulated by Arno van den Essen in June 1994, which can be found in [11]. Hethen used a computer algebra system to verify a human guided reduction path to aYagzhev map in dimension 203, then explicitly computed a Gorni-Zampieri pairing[12] to a Dru˙zkowski map in dimension 1999, using sparse matrix representationsas necessary. These details are excerpted from a comprehensive unpublished noteby Hubbers, which he made available.Remark. Dru˙zkowski obviously did not use GZ pairing, since it was unknown atthe time. But it also preserves the same two key properties in the SRJC context.More recently, reductions of the ordinary JC to the symmetric case have beenconsidered, primarily over R and C . Let k denote a field of characteristic zero. Inthe JC world a polynomial map F : k n → k n is often called symmetric, in a startlingabuse of language, if J ( F ) is a symmetric matrix. In that case, F is the gradientmap of a polynomial function h : k n → k and J ( F ) is the Hessian matrix of secondorder partial derivatives of h . So in the symmetric case, the JC becomes the Hessianconjecture (HC), namely that gradient maps of polynomials with constant nonzero EDUCTION THEOREMS FOR THE STRONG REAL JACOBIAN CONJECTURE 3
Hessian determinant have polynomial inverses. In [17], Guowu Meng proves theequivalence of the JC and the HC, using what he refers to as a trick. Meng’s trickreplaces a map F = ( f , . . . , f n ) in the variables x , . . . , x n by the map in the 2 n variables y , . . . , y n , x , . . . , x n obtained by taking the gradient of the scalar function y f + · · · + y n f n . For k = R , this construction works even for twice continuouslydifferentiable maps. In the SRJC context it provides a one step reduction to thesymmetric case that also preserves the Keller property in both directions.In [2], Michiel de Bondt and Arno van den Essen prove a more targeted reductionover C , namely to symmetric Keller Yagzhev maps. The reduction process involvesthe use of √−
1, and if applied to a real Keller map may yield a Yagzhev map thatis not real. Interestingly, it has been shownthat all complex symmetric Keller Dru˙zkowski maps have polynomial inverses[3, 9]. 3.
Stable and Segre equivalence
Two maps, F and G , from a topological space A to another one B , are calledtopologically equivalent if F = h B ◦ G ◦ h A , where h A and h B are homeomorphisms,respectively of A to itself and of B to itself. In other words, F and G are the samemap up to coordinate changes in the domain and codomain by topological auto-morphisms. Topological stable equivalence for the set of all maps F : R n → R n in all dimensions n > F = ( f , . . . , f n ), and its exten-sion by fresh variables to G = ( f , . . . , f n , x n +1 , . . . , x m ) for any m > n . Thereare many other types of stable equivalence, such as real analytic or polynomial,each characterized by the type of automorphisms allowed for (global) coordinatechanges. Stable equivalence, unqualified, will refer to the least restrictive, purelyset theoretic, type, with all bijections allowed as automorphisms.For brevity, call F : R n → R n (1) nondegenerate if j ( F ) is not identically zero,(2) nonsingular if j ( F ) (cid:54) = 0 everywhere, and (3) a Keller map if j ( F ) is a nonzeroconstant. These terms are meant to imply that J ( F ), the Jacobian matrix of F ,exists at every point of R n , and can be applied to any such F if the correspondingrestriction on j ( F ) is satisfied. For polynomial stable equivalence, the applicableautomorphisms are polynomial maps with polynomial inverses, making it obviousthat such equivalence preserves each of the above three properties. All preserva-tion properties in this paper apply equally well in both directions. It also clearlypreserves each of the properties of being everywhere defined rational, polynomial,injective, surjective, or bijective. If the maps are nonsingular, it preserves theexistence of a rational or polynomial inverse.To verify this last assertion in the case of extension by fresh variables, one checksthe Jacobian matrices to see that the appropriate part of an inverse in the largernumber of variables is independent of the fresh variables and restricts to an inversein the smaller number of variables.The slightly more general and less familiar concept of birational stable equiva-lence allows the use of automorphisms that are everywhere defined rational mapswith everywhere defined rational inverses. By inspection of the arguments in thepolynomial case, one sees easily that birational stable equivalence has all the preser-vation properties listed above for the polynomial case, except that it need notpreserve polynomial maps, polynomial inverses, or the Keller property. L. ANDREW CAMPBELL If F = ( f , . . . , f n ) is an everywhere defined rational map and is nondegenerate,then its components are algebraically independent over R in the field R ( X ) ofrational functions in the coordinate variables X = x , . . . , x n , and so they generate asubfield R ( F ) ⊆ R ( X ) over R , that is also a rational function field in n variables over R . Even without nonsingularity, the extension R ( X ) / R ( F ) permits the definitionof dex and sag as in the introduction. The extension degree d = dex is finite andequal to the degree of the minimal polynomial over R ( F ) of any h ∈ R ( X ) that isprimitive, meaning that h generates R ( X ) as a field over R ( F ). For such an h , thepowers h i for i = 0 , . . . , d − R ( X ) as a vector space over R ( F ).An automorphism of the extension is, by definition, a field automorphism of R ( X )that fixes every element of R ( F ). So it is linear over R ( F ), and a multiplicativehomomorphism, hence completely determined by its value on h . That value mustbe a root of the minimal polynomial of h and must lie in R ( X ), and any such rootdetermines a unique automorphism of the extension. Thus sag is the number ofsuch roots, which is therefore the same for any choice of h .In another relaxation of assumptions, it suffices to assume that F is an every-where defined nondegenerate rational map and an open map in order to concludethat it is quasifinite and that the maximum fiber size is at most dex.The main concern here is the case of everywhere defined rational nonsingularmaps, for which all the assertions in the introduction are proved in [7]. Theorem 1.
Assume F : R n → R n and G : R m → R m are birationally stablyequivalent. If either one is an everywhere defined rational nonsingular map, thenso is the other, and each of the numerical attributes dex, sag, and mfs has the samevalue for both maps.Proof. Only the equality of the numerical attributes needs checking. And it needsto be checked only for the generating equivalences.Suppose first that m > n and G = ( F, Z ), with Z a list of fresh variables. Forany y ∈ R n and z ∈ R m − n , the fiber of F over y is the same size as the fiber of G over ( y, z ), so mfs is preserved. A primitive element for R ( X ) / R ( F ) is clearlyalso a primitive element for R ( X, Z ) / R ( G ). Since the tensor product over R with R ( Z ) is an exact functor, the associated power basis for the first extension is alsoone for the second, and so dex is preserved. An automorphism of the extension R ( X, Z ) / R ( G ) is determined by the image of the primitive element. That image isa root of the minimal polynomial for the primitive element. That polynomial hascoefficients independent of the fresh variables in Z . On a Zariski open subset of R m ,where the root is a real analytic function of the coefficients, the root is independentof the fresh variables. So the first order partials with respect to those variables areidentically zero and the root lies in R ( X ). Thus the automorphism is uniquely thenatural lift of an automorphism of R ( X ) / R ( F ), and so sag is preserved.Second, suppose that m = n and G = A ◦ F ◦ B , with A and B everywheredefined birational automorphisms. Viewing them as coordinate changes makes itclear that mfs is preserved. It suffices to consider further only the special cases i) G = A ◦ F and ii) G = F ◦ B , and A , B , and their inverses do not need to bedefined everywhere.In case i) A induces an automorphism of R ( F ), so R ( F ) and R ( A ◦ F ) are thesame subfield of R ( X ), hence the two extensions are the same and so have the sameproperties. In case ii) the two extensions are R ( X ) / R ( F ) and R ( X ) / R ( F ◦ B ),which are generally different extensions. Take a primitive element h for the first EDUCTION THEOREMS FOR THE STRONG REAL JACOBIAN CONJECTURE 5 extension, then apply the automorphism of R ( X ) induced by B to h , its minimalpolynomial over R ( F ), and the roots of that polynomial in R ( X ). The image of h is primitive over R ( F ◦ B ), the new polynomial is irreducible there, and the rootsof the two polynomials correspond. So dex and sag are preserved. (cid:3) Let F : R n → R n be a polynomial map satisfying F (0) = 0. Then H ( x, t ) =(1 /t ) F ( tx ) is polynomial and provides, for 0 ≤ t ≤
1, a homotopy between thelinear part of F and F itself.This Segre homotopy [21] can be generalized in many ways, e.g. to the case ofa complex map or parameter t and to analytic or rational maps, not to mentionformal and convergent power series. It is used here to define the concept of Segreequivalence on the set of real analytic endomorphisms of R n ( n >
0) that fix 0.It is the equivalence relation generated by declaring F : R n → R n equivalent to G : R n +1 → R n +1 , with G ( x, t ) = ( F ( tx ) /t, t ). In that case, for t (cid:54) = 0 considerationof the Jacobian matrix of G shows that j ( G )( x, t ) = j ( F )( tx ), a result that then alsoholds for t = 0, by continuity. So Segre equivalence preserves nondegeneracy, non-singularity, and the Keller property. Again all preservation properties apply in bothdirections. It also preserves polynomial maps and everywhere defined rational maps,because G ( x,
1) = ( F ( x ) , t (cid:54) = 0, the set G − ( y, t ) = { ( x/t, t ) | x ∈ F − ( ty ) } for any y ∈ R n . That implies that injectivity and surjectivity are preserved providedthat G is bijective on the set of points ( x, j ( F )(0) (cid:54) = 0.In particular, for bijective nonsingular F one has G − ( y, t ) = ( F − ( ty ) /t, t ), andso polynomial and everywhere defined rational inverses are also preserved. Theorem 2.
Assume F : R n → R n and G : R m → R m are real analytic mapssending to and that they are Segre equivalent. If either one is an everywheredefined rational nonsingular map, then so is the other, and each of the numericalattributes dex, sag, and mfs has the same value for both maps.Proof. Only the equality of the attributes needs checking, and only for the case G ( x, t ) = ( F ( tx ) /t, t ). Fibers over points ( y,
0) are of size 1 and for t (cid:54) = 0 the fiberof G over ( y, t ) has the same size as the fiber of F over ty , by the formula givenabove for the set G − ( y, t ). So mfs is preserved.Now consider the automorphism of the field R ( X, t ) that sends x i to tx i and t to itself. It restricts to an isomorphism of R ( F, t ) onto R ( G ).This is just an instance of special case ii) in the proof of the previous theorem.So dex and sag have the same value for G as for ( F, t ), and hence, by the previoustheorem, as for F . (cid:3) The coimage of a map F : R n → R n is the set R n \ F ( R n ) of points in thecodomain that are not in the image of F . In the complex JC context, it is well knownthat the coimage has complex codimension at least two. Briefly, the reasoning isas follows. Since the coimage is closed and constructible, if it has codimension lessthan two it contains an irreducible hypersurface h = 0, h ◦ F vanishes nowhere andso is constant, contradicting the algebraic independence of the components of F . Inthe SRJC and RRJC contexts, there are no parallel results for the real codimensionof the coimage, even if the map has dense image. The Pinchuk maps, however, dohave finite coimages, which are indeed of codimension two in R . L. ANDREW CAMPBELL
Theorem 3.
Assume F : R n → R n and G : R m → R m are everywhere definedrational nonsingular maps and that they are birationally stably or Segre equivalent.Then the codimension of the coimage is the same for both maps.Proof. Only the Segre case is not totally trivial, and in the base Segre case thecoimage of G consists of the point ( a/t, t ) for a in the coimage of F and t (cid:54) = 0. (cid:3) Gorni-Zampieri pairing
Let f i = x i + l i be the components of a map F of cubic linear type in dimension n . It is customary to write F in the compact form F ( x ) = x +( Ax ) ∗ , where A is thematrix of coefficients of the linear forms and the exponent indicates componentwisecubing. Let G be a map of cubic homogeneous type in dimension m < n . A GZpairing between G and F is given by two matrices B and C , respectively of sizes m × n and n × m , satisfying BC = I , ker B = ker A , and G ( x ) = BF ( Cx ) forall x ∈ R m . The original definition [12] writes F as F ( x ) = x − ( Ax ) ∗ , but thedifferent sign affects only some formulas not used here. Theorem 4. If G and F are GZ paired, then they are polynomially stably equiva-lent.Proof. Note that ker C = 0, Im B = R m , and that R n is the direct sum of Im C andker B . Choose a linear isomorphism D from R n − m to ker B . Let E be its inverse.Consider the extension of G by fresh variables to G (cid:48) = ( g , . . . , g m , z m +1 , . . . , z n ).Let C (cid:48) ( x, z ) = Cx + D ( z ) ∈ R n and B (cid:48) ( x ) = ( Bx, E (cid:48) ( x )), where E (cid:48) is the linearextension of E to R n that is 0 on Im C .Note that both B (cid:48) and C (cid:48) are linear automorphisms of R n . Observe that F ( Cx + D ( z )) = Cx + D ( z ) + ( ACx ) ∗ . So ( B (cid:48) ◦ F ◦ C (cid:48) )( x, z ) = ( G ( x ) , z + E (cid:48) (( ACx ) ∗ )) = G (cid:48) ◦ ( x, z + H ( x )), where H is cubic homogeneous. Since ( x, z + H ( x )) has theobvious inverse ( x, z − H ( x )), it follows that G and F are polynomially stablyequivalent. (cid:3) Remarks. The same reasoning works over any ground field k . There is alsonothing special about the use of 3 as the exponent. All works just as well for powerhomogeneous and power linear maps of the same degree d > A , is the same as the dimension m of the map of cubichomogeneous type. In [10] the SRJC is proved for all maps of cubic linear type andrank 2. The heart of the proof is a theorem proving that the SRJC is true for allmaps of cubic homogeneous type in dimension 2. These facts are of interest whenconsidering structured counterexamples in higher dimensions.Remarks. For reasons not clear to me, [10] presents the results mentioned abovefor maps with an everywhere positive Jacobian determinant, which is automaticallytrue for nonsingular maps of cubic homogeneous type. Other results include theSRJC for all maps of cubic linear type in dimension 3.The dimension 2 results were later improved to cover polynomial maps with com-ponents of degree at most 3 [13], and then to polynomial maps with one componentof degree at most 3 [4]. 5. Main results
Theorem 5.
There is an algorithm that transforms a nondegenerate, polynomialmap F : R n → R n into a map G : R m → R m of cubic homogeneous type, where m EDUCTION THEOREMS FOR THE STRONG REAL JACOBIAN CONJECTURE 7 is generally much larger than n , using polynomially stable equivalences and a singleSegre equivalence.Proof. In each step below a map F is replaced by a map G , which becomes thenew F for the next step. At each step both F and G are nondegenerate, since thatproperty is preserved by the equivalences.Step 1. Lower the degree. Suppose F = ( f , . . . , f n ). F is polynomially stablyequivalent to ( f − ( y + a )( z + b ) , f , . . . , f n , y + a, z + b ), where a, b are polynomialsthat depend only on x , . . . , x n . Thus, if a term of f has the form ab , with deg( a ) > b ) >
1, it can be removed at the cost of introducing two new variablesand some terms of degree less than deg( ab ). Repeating this for terms of maximumdegree until there are no more maximal degree terms of the specified form in anycomponent, one finally obtains a polynomial map G (in a generally much higherdimension), all of whose terms are of degree no more than three. This is a standardalgorithm [1, 11]. There is flexibility in the choice of term to remove next, and onecan opportunistically remove a product ab that is not a single term, making choicesto reach a cubic map more quickly. This step is a polynomial stable equivalence.Step 2. Normalize. F is now cubic and (still) nondegenerate. Let n be thecurrent dimension. Choose x ∈ R n with j ( F )( x ) (cid:54) = 0. After suitable translations,( J ( F )( x )) − F becomes a cubic map G , such that G (0) = 0 and G (cid:48) (0) = J ( G )(0) isthe identity matrix I . This step is an affine (in the vector space sense) equivalence.Step 3. Segre equivalence. Now F = X + Q + C , where Q and C are, respectively,the quadratic and cubic homogeneous components of F . Let t be a new variable,and put G = ( X + tQ + t C, t ). This is a polynomial Segre equivalence, as definedpreviously.Step 4. Final step. Now F = ( X + tQ + t C, t ), with Q quadratic homogeneousand C cubic homogeneous, and both independent of t . Define two polynomialautomorphisms A , A in X, Y, t , where Y is a sequence of n additional variables,by A = ( X − t Y, Y, t ) and A = ( X, Y + C, t ). Then G = A ◦ ( X + tQ + t C, Y, t ) ◦ A is the map of cubic homogeneous type ( X − t Y + tQ, Y + C, t ). This step is apolynomial stable equivalence. (cid:3)
The theorem and proof are valid over C as well as over R , and, indeed, moregenerally for Keller maps. All proofs of reduction to cubic homogeneous type startwith reduction to degree 3, followed by elimination of the quadratic terms. Thegiven proof most closely follows that of Dru˙zkowski in [8], which explicitly allowsfor nonconstant Jacobian determinants.The main point of the given proof is that for nonsingular polynomial maps, by thepreservation results previously proved, the reduction preserves (in both directions)not only bijectivity, but also dex, mfs, sag, the Keller property, and the codimensionof the coimage.So if it is applied to a Pinchuk map, it yields a Yagzhev map G , for which j ( G ) isnot constant and J ( G ) is not unipotent. Up to inessential details, Hubbers followsthe above steps in the first part of his 1999 reduction, obtaining a cubic mapin dimension n = 101 and then a map of cubic homogeneous type in dimension2 n + 1 = 203. Hubbers’ Yagzhev map in dimension 203 is thus not Keller andsatisfies dex = 6, mfs = 2, sag = 1, and has a coimage of codimension 2.A further reduction to a map of cubic linear type can be effected using themethod of Dru˙zkowski in [8] or the method of GZ pairing developed by GianlucaGorni and Gaetano Zampieri in [12]. Since GZ pairing has been shown to be a L. ANDREW CAMPBELL polynomial stable equivalence, Hubbers’ final Dru˙zkowski map in dimension 1999has the same properties as those stated for his Yagzhev map in dimension 203.
Theorem 6.
Any nonsingular C map F : R n → R n , is stably equivalent to a C nonsingular map G : R n → R n with a symmetric Jacobian matrix. The equiv-alence is birationally stable if F is everywhere defined rational, and polynomiallystable if F is a polynomial Keller map.Proof. This is Meng’s trick with a reordering of the variables. Let z = ( x, v ) beany point of R n , with x = ( x , . . . , x n ) and v = ( x n +1 , . . . , x n ). Let h = x · F ( v ),where the dot denotes the standard inner product of n -vectors. Let G be thegradient of the scalar function h . Then G has a symmetric Jacobian matrix and G = ( F ( v ) , x · J ( F )( v )), with the dot now denoting a vector matrix product and J ( F ) the Jacobian matrix of F . But ( F ( v ) , x · J ( F )( v )) = ( F ( x ) , v · J ( F )( x )) ◦ ( v, x )and ( F ( x ) , v · J ( F )( x )) = ( F ( x ) , v ) ◦ ( x, v · J ( F )( x )). Since ( x, v · J ( F )( x )) has the C inverse ( x, v · J ( F ) − ( x )), the composition A = ( x, v · J ( F )( x )) ◦ ( v, x ) is a C automorphism. Moreover, if F is an everywhere defined rational or polynomialKeller map, it is clear that A has the claimed properties. (cid:3) This theorem reduces the entire RRJC, not just the SRJC, to the case of asymmetric Jacobian matrix and preserves dex, sag, and mfs.It is natural to attempt to combine the two main results by applying Theorem 6to a Yagzhev map. The resulting map is polynomial, with only its linear and cubichomogeneous components nonzero. But its linear part is not the identity.6.
Acknowledgments
Thanks especially to Engelbert Hubbers for providing complete details on hisreduction procedure [14]. Michiel deBondt helped simplify the last step of theproof of Theorem 5. And both he and Gianluca Gorni sent me proofs that GZpairing preserves mfs, when I first raised the question.
References [1] Hyman Bass, Edwin H. Connell, and David Wright. The Jacobian conjecture: reduction ofdegree and formal expansion of the inverse.
Bull. Amer. Math. Soc. (N.S.) , 7(2):287–330,1982.[2] Michiel de Bondt and Arno van den Essen. A reduction of the Jacobian conjecture to thesymmetric case.
Proc. Amer. Math. Soc. , 133(8):2201–2205 (electronic), 2005.[3] Michiel de Bondt and Arno van den Essen. The Jacobian conjecture for symmetric Dru˙zkowskimappings.
Ann. Polon. Math. , 86(1):43–46, 2005.[4] Francisco Braun and Jos´e Ruidival dos Santos Filho. The real Jacobian conjecture on R istrue when one of the components has degree 3. Discrete Contin. Dyn. Syst. , 26(1):75–87,2010.[5] L. Andrew Campbell. The asymptotic variety of a Pinchuk map as a polynomial curve.
Appl.Math. Lett. , 24(1):62–65, 2011.[6] L. Andrew Campbell. Pinchuk maps and function fields.
J. Pure Appl. Algebra , 218(2):297–302, 2014.[7] L. Andrew Campbell. On the rational real Jacobian conjecture.
Univ. Iagel. Acta Math. , toappear, 2014. ArXiv 1210.0251.[8] Ludwik M. Dru˙zkowski. An effective approach to Keller’s Jacobian conjecture.
Math. Ann. ,264(3):303–313, 1983.[9] Ludwik M. Dru˙zkowski. The Jacobian conjecture: symmetric reduction and solution in thesymmetric cubic linear case.
Ann. Polon. Math. , 87:83–92, 2005.
EDUCTION THEOREMS FOR THE STRONG REAL JACOBIAN CONJECTURE 9 [10] Ludwik M. Dru˙zkowski and Kamil Rusek. The real Jacobian conjecture for cubic linear mapsof rank two.
Univ. Iagel. Acta Math. , 32:17–23, 1995.[11] Arno van den Essen.
Polynomial automorphisms and the Jacobian conjecture , volume 190 of
Progress in Mathematics . Birkh¨auser Verlag, Basel, 2000.[12] Gianluca Gorni and Gaetano Zampieri. On cubic-linear polynomial mappings.
Indag. Math.(N.S.) , 8(4):471–492, 1997.[13] Janusz Gwo´zdziewicz. The real Jacobian conjecture for polynomials of degree 3.
Ann. Polon.Math. , 76(1-2):121–125, 2001. Polynomial automorphisms and related topics (Krak´ow, 1999).[14] Engelbert Hubbers. Pinchuk’s 2-dimensional example paired to a cubic linear 1999-dimensional map. Preprint dated November 6, 1999. Personal communication, 2010.[15] A. V. Jagˇzev. On a problem of O.-H. Keller.
Sibirsk. Mat. Zh. , 21(5):141–150, 191, 1980.[16] O. H. Keller. Ganze Cremona-Transformationen.
Monatshefte der Mathematischen Physik ,47:299–306, 1939.[17] Guowu Meng. Legendre transform, Hessian conjecture and tree formula.
Appl. Math. Lett. ,19(6):503–510, 2006.[18] T. Parthasarathy.
On Global Univalence Theorems , volume 977 of
Lecture Notes in Mathe-matics . Springer Verlag, New York, 1983.[19] Sergey Pinchuk. A counterexample to the strong real Jacobian conjecture.
Math. Z. , 217(1):1–4, 1994.[20] John D. Randall. The real Jacobian problem. In
Proc. Sympos. Pure Math. 40 , pages 411–414,Providence, RI, 1983. American Mathematical Society.[21] Beniamino Segre. Variazione continua ed omotopia in geometria algebrica.
Ann. Mat. PuraAppl. (4) , 50:149–186, 1960.
908 Fire Dance Lane, Palm Desert CA 92211, USA
E-mail address ::