arXiv:2002.10249v4 [math.GM] 17 May 2020 n sas oyoilmp u oOgo’ hoe,teJac the Theorem, Osgood’s to Due map. for a also is and sanneocntn in constant nonzero a is a 9 2020 19, May odto.Aplnma map polynomial A Condition. variables ensampfrom map a defines fteprildrvtvsof derivatives partial the of Let P map h inverse the n13 for 1939 in hssuyi on opeetapoffrtecnetr.Our conjecture. uses the additionally for and proof Theorem, a Diffeomorphism Hadamard’s present to going is study This en natmrhs.Kle 3]pooe h aoinCo Jacobian the proposed [35] Keller automorphism. an being K nOtmzto praht Jacobian to Approach Optimization An Let -a,Jcba ojcue oyoilAtmrhs,Pro Automorphism, Polynomial Conjecture, Jacobian D-map, P eoeanme ed nwhich on field, number a denote X from n det ( = ≥ hnqn nttt fGenadItlietTechnology Intelligent and Green of Institute Chongqing P K ( 2 − K x J n and 1 1 n P = x , . . . , xssadi loaplnma a.Ti ojcuewsfirst was conjecture This map. polynomial a also is and exists ) K oisl,i h eemnn fisJcba arxi nonz a is matrix Jacobian its of determinant the if itself, to K C of n 2 eanme edo characteristic of field number a be to J n u nSaes19 ito ahmtclPolm o the for Problems Mathematical of list 1998 Smale’s in put and n P K K ) ahplnma vector polynomial Each . n A . saplnma ucinof function polynomial a is P The . j hns cdm fSciences of Academy Chinese ∈ P ihrsetto respect with ∈ P

n K × [email protected] : Conjecture [ .I I. K n X [ aoinmatrix Jacobian ] X n NTRODUCTION ne Terms Index in Liu Jiang ] san is n K Abstract [ scalled is X x = ] k automorphism htis, that , 0 aoinCnetr set o polynomial a for asserts Conjecture Jacobian . K P [ x elrmap Keller ( X oeotmzto idea. optimization some 1 J X ro sbsdo r˙kwk a and Dru˙zkowski Map on based is proof x , . . . , P . J =( := ) ( e Map per aoinCondition Jacobian P X [ = ) of n ( J ] P ∂x ∂P P ba odto snecessary is Condition obian K eteplnma igin ring polynomial the be 1 k j 2]i tstse Jacobian satisfies it if [28] ( jcue( njecture o hr)of short) for n ,k j, , X fteinverse the if r osatin constant ero ) ypooe yKeller by proposed ly , . . . , 1 = P n , . . . , C J etCentury. Next n ( sthat is P for ) X ( )) X ] K P hnthe Then . ) ∈ K − then det consists 1 n K exists DRAFT = ( [ X J C P ] n n 2 ) 1 2 in 1939, where C is the complexity field. We refer the reader to [7], [20], [28], [29], [54] for a nice survey paper containing some history and updated progresses of the Jacobian Conjecture. Abhyankar [2] gave its modern style as follows. Jacobian Conjecture — (K, n): If K[X]n is a Keller map where K is an arbitrary J C P ∈ field of characteristic 0 and the integer n 2, then is an automorphism. ≥ P (K, 1) is trivially true. Besides, there are simple counterexamples [3], [7] for (K, n) J C J C when the number field K has characteristic > 0. So the characteristic 0 condition is necessary. These two assumptions are always supposed to be true in the rest of this article unless a related statement is specified particularly. Let (K) stand for (K, n) being true for all positive J C J C integers n. Due to Lefschetz Principle, it suffices to deal only with (C). For polynomial J C maps in C[X]n, it is well known that Theorem 1.1 ( [7], [43], [52], [55]): If a polynomial map P C[X]n is injective then it must ∈ be an automorphism. Thus, it suffices to show that each Keller map in C[X]n is injective. There is a remarkable result [51] for polynomial maps of degree 2. Theorem 1.2: If (X) K[X]n is a quadratic polynomial, then (X) is injective iff for each P ∈ P X K it alway has det(J (X)) =0. ∈ P 6 Proof: For the result, we present the wonderful proof gave in [24]. The assertion is true by X + Y (X) (Y )= J (X Y ) (1) P − P P 2 −  

For high degree polynomial maps, Yagˇzev [55] and independently Bass et al. [7] showed that it is sufficient to consider Keller maps of form (X) := X + (X) C[X]n which is P H ∈ called Yagzevˇ map, where (X) is a homogeneous polynomial vectors of degree 3. Based on H this result, Dru˙zkowski [21] further showed that it suffices to consider Keller maps of so-called cubic linear form (X) := X +( X)∗3 C[X]n, nowadays, called Druzkowski˙ map (D-map P A ∈ for short), wherein ( X)∗3 be the vector whose k-th element is (AT X)3 and AT is the k-th A k k row vector of . That is, if each D-map (X) C[X]n is injective then Jacobian Conjecture is A P ∈ true. It is natural to ask whether all D-map (X) R[X]n being injective also implies Jacobian P ∈ Conjecture. The answer is yes in what follows. Theorem 1.3: If each D-map in R[X]n is injective for all n, then each Keller map P C[X]n ∈ is injective and so an automorphism.

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Proof: Assume variable X := Y + iZ with i = √ 1, that is, xk = yk + izk for 1 k n. − ≤ ≤ Like the argument on the first page in [22] or in the proof of Lemma 6.3.13 on page 130 in [28], n ⋆ a D-map P(X) := P(Y + iZ) C[X] can be treated as a real map P (Y,Z) := (ReP1(Y + ∈ 2n iZ),ImP1(Y + iZ), ,RePn(Y + iZ),ImPn(Y + iZ)) R[Y,Z] , where RePk(Y + iZ) ··· ∈ and ImPk(Y + iZ) are the real and imaginary parts of Pk(Y + iZ), respectively. It is evident that P(Y + iZ) C[X]n and P⋆(Y,Z) R[Y,Z]2n have the same injectivity. In fact, each ∈ ∈ ∗ RePk(Y +iZ) and ImPk(Y +iZ) for 1 k n are of forms Yk +Hk(Y,Z) and Zk +H (Y,Z), ≤ ≤ k ∗ respectively. Furthermore, both Hk(Y,Z) and Hk (Y,Z) are homogeneous polynomial about variables (Y,Z) of degree 3, which can be seen by expanding the k-th component of D-map 2 Pk(Y + iZ). It is well known det(JP⋆ (Y,Z)) = det(JP(Y + iZ)) . As P(Y + iZ) is a D-map, | | ⋆ det(JP⋆ (Y,Z))=1 and thus P (Y,Z) is actually a Yagˇzev map. Now, we apply Dru˙zkowski’s reduction in [21] to get that there is some D-map P@(Y,Z, Z˜) R[Y,Z, Z˜]m with m n such ∈ ≥ that P⋆(Y,Z) is injective iff P@(Y,Z, Z˜) is injective, by Theorem 3 and Remark 4 in [21]. In consequence, if each D-map in R[X]n is injective for all n, then each Keller map P C[X]n ∈ is injective and so an automorphism. As a result of Theorem 1.3, Jacobian Conjecture is true if each D-map in R[X]n is injective for all n. In the study, we will use Hadamard’s Diffeomorphism Theorem to show the injectivity of D-maps in R[X]n. By Jacobian Condition, each Keller map is a local homeomorphism. Even for analytic maps, Hadamard’s Diffeomorphism Theorem presents a necessary and sufficient condition for local homeomorphism being diffeomorphisme in terms of the proper map. A differentiable map : Rn Rn is an diffeomorphism if it is bijective and its inverse is F 7→ also differentiable. A continuous map : Rn Rn is proper if −1(C) is compact for each H 7→ H compact set C Rn. Equivalently, a map is proper iff it maps each unbounded set B into ⊂ H an unbounded set (B). Based on these notions, in ( [33], [34]) it was showed that H Hadamard Theorem 1.4: For an analytic map : Rn Rn, is a diffeomorphism iff H 7→ H det J (X) =0 for all X Rn and is proper. H 6 ∈ H Therefore, it suffices to show the properness of D-maps in R[X]n for their injectivity. To this end, we are going to show the following crucial result in the next section. Proper Map Theorem 1.5: Each D-map (X) := X +( X)∗3 in R[X]n is proper and so D A injective. The basic idea of its proof is in what follows. Firstly, we show +λ ( X)∆2 is invertible for I A A each λ R, where is the identity matrix. Then we show the result by contradiction. Roughly, ∈ I

May 19, 2020 DRAFT 4 if (X) is not proper, then we will obtain a point Z = Y such that Z + ℓ Z∗3 = 0 for some D A A ℓ R and Y Rn. The detail will be depicted in the next section. By Theorem 1.3 we infer ∈ ∈ Polynomial Automorphism Theorem 1.6: For every n 2 and any field K of characteristic 0, ≥ each Keller map K[X]n is an automorphism. P ∈ In the rest of this article, we assume n 2 for the dimension of Rn unless it is specified ≥ in particular. For clarity, let capital letters denote vectors, blackboard bold letters denote sets of vectors (points), fraktur letters denote vector functions, little letters denote integers, Greek letters denote real numbers and real functions.

II. PROOF OF PROPER MAP THEOREM

Let’s start with some basic notions and notations. A set B Rn is called unbunded if for ⊆ any γ R there is an element X B such that X >γ, where is the Euclidean norm, ∈ ∈ || || ||·|| 2 U Rn Rn that is, X := Xk . Let := X X = 1 , which is a compact set in . || || 1≤k≤n { ∈ | || || } Given a matrix r, letP T denote its transpose, define Im( ) := X X Rn . Given a A A A {A | ∈ } T ∆2 2 2 vector V = (v1, , vn) , let V denote the diagonal matrix whose diagonals are v ,...,v . ··· 1 n For a D-map (X) := X +( X)∗3, its Jacobian matrix is J = + 3( X)∆2 . From [21], D A P I A A Theorem 2.1: (X) := X +( X)∗3 is a D-map iff ( X)∆2 is nilpotent matrix for any X. D A A A From this theorem, the following proposition is evident. ∗3 Proposition 2.2: If (X) := X +( X) is a D-map, then for all λ R, the map λ(X) := D A ∈ D X + λ( X)∗3 is also a D-map. Thus, λ( X)∆2 and so λ ( X)∆2 are nilpotent matrices. A A A A A Therefore, + λ ( X)∆2 is invertible. I A A Proof: By Theorem 2.1, if (X) := X +( X)∗3 is a D-map, then ( X)∆2 is nilpotent D A A A ∆2 k k+1 matrix, and so is λ( X) for any λ R. It is evident that ( ) =0n n implies ( ) = A A ∈ AB × BA k n×n ( ) = 0n n for any integer k and arbitrary matrices , R , where 0n n is the B AB A × A B ∈ × zero matrix of size n n. Hence, all λ ( X)∆2 are nilpotent matrices. In consequence, each × A A + λ ( X)∆2 is invertible. I A A

This result accomplished the first part of the proof for Theorem 1.5. To finish the proof, we are going to show Lemma 2.3: If a linear cubic form ˆ(X) := X +( X)∗3 is not proper, then (I) there is an C A T ∗3 unbounded Ui Im( ) i such that Ui + U i is bounded, that is, ˆ(X) is not proper on { ∈ AA } { A i } C T T ∗3 Im( ); and (II) there is some W Im( ) such that ( W ) =0n n but W =0n n. AA ∈ A A A × A 6 ×

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To show this result, we need following preliminary materials. Given a matrix Rn×n, let A ∈ A⊥ := X Rn X = 0 denote the linear space in which each element is a solution of { ∈ | A } X = 0. For any X A⊥ and Y = T Z Im( T ), we have Y T X = ZT X = 0, that is, A ∈ A ∈ A A Y X. In consequence, ⊥ Proposition 2.4: The spaces A⊥ and Im( T ) have the following properties: A 1) A⊥ and Im( T ) are orthogonal to each other. A 2) A⊥ Im( T )= 0, and so Y =0 for any nonzero Y Im( T ). ∩ A ||A || 6 ∈ A 3) Both A⊥ and Im( T ) are closed sets. A 4) For any nonzero Z Rn, it can be uniquely decomposed as Z = X + Y such that X Y , ∈ ⊥ X A⊥ and Y Im( T ). ∈ ∈ A Let’s set U(A⊥) := U A⊥ and U(Im( T )) := U Im( T ), then ∩ A ∩ A Proposition 2.5: Both U(A⊥) and U(Im( T )) are closed and so compact. Moreover, Y = A ||A || 6 0 for any Y U(Im( T )). ∈ A Now, we present the proof of Lemma 2.3 in what follows.

Proof: If ˆ(X) is not proper, then there must be an unbounded sequence Yi i such that C { } (Yi) i is a bounded sequence. Without loss of generality, we suppose that Yi+1 > Yi for {C } || || || || all i. By the boundness of (Yi) i, there is some real number σ < such that {C } ∞ T T ∗3 ∗3 2 2 Y Yi +2Y ( Yi) + ( Yi) σ (2) i i A || A || ≤ T ∗3 for all i. Because Yi i is unbounded and and (Yi) i is bounded, it must be Y ( Yi) < { } {C } i A T ∗3 0 for almost all i. Without loss of generality, we suppose Y ( Yi) < 0 for all i. As a i A ∗3 ∗3 result, ( Yi) = 0 and so ( Yi) > 0 for all i. Then by optimization theory on quadratic A 6 || A || [11], we have (Y T ( Y )∗3)2 Y ∗3 2γ2 Y T Y ∗3γ Y T Y Y T Y i i (3) ( i) +2 i ( i) + i i i i A ∗3 2 || A || A ≥ − ( Yi) || A || for any γ R and each i. When γ =1, we get ∈ (Y T ( Y )∗3)2 Y T Y i i Y T Y Y T Y ∗3 Y ∗3 2 σ2 (4) i i A ∗3 2 i i +2 i ( i) + ( i) − ( Yi) ≤ A || A || ≤ || A || Therefore, (Y T ( Y )∗3)2 σ2 i i (5) lim 1 T A ∗3 2 lim T =0 i→∞ − Y Yi ( Yi) ≤ n→∞ Y Yi  i || A ||  i Yi ∗3 since Yi i is unbounded. Let Xi = , then Xi U, ( Xi) = 0 by the assumption { } ||Yi|| ∈ A 6 T ∗3 Y ( Yi) < 0 for all i, and then i A

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Corollary 2.6: XT ( X )∗3 Y T ( Y )∗3 i i i i (6) lim A ∗3 = lim A ∗3 = 1 i→∞ ( Xi) i→∞ Yi ( Yi) − || A || || ||·|| A || 2 ⊥ For each i, we decompose Xi as Xi = αiVi + 1 α Wi such that Vi U(A ), Wi − i ∈ ∈ T T ∗3 U(Im( )), and αi [0, 1]. By assumption Y ( Ypi) < 0 for all i, we have A ∈ i A Corollary 2.7: αi < 1 for all i.

Because Vi, Wi U for all i, we can choose a subsequence Xi k of Xi i such that ∈ { k } { } n n ⊥ T lim Vi = V∞ R and lim Wi = W∞ R . As U(A ) and U(Im( )) are closed, it must k→∞ k ∈ k→∞ k ∈ A ⊥ T be V U(A ) and W U(Im( )). Furthermore, we can choose a subsequence Xik j ∞ ∈ ∞ ∈ A { j }

of Xi k such that lim αikj = α∞ [0, 1]. Without loss of generality, we assume Xi i has { k } j→∞ ∈ { } such properties of Xik j, that is, { j } (i) lim αi = α∞ [0, 1]. i→∞ ∈ ⊥ T (ii) lim Vi = V∞ U(A ) and lim Wi = W∞ U(Im( )) i→∞ ∈ i→∞ ∈ A Then by the decomposition of Xi, each Yi can be uniquely decomposed as Yi = αi Yi Vi + || || 2 2 1 α Yi Wi. If 1 α Yi Wi i is bounded, then αi Yi Vi i and so (Yi)= αi Yi Vi+ − i || || { − i || || } { || || } {C || || 2 2 3 ∗3 p1 α Yi Wi +( p1 α Yi ) ( Wi) i must be unbounded since Yi i is unbounded. − i || || − i || || A } { } 2 2 Therefore,p 1 α pYi Wi i and so 1 α Yi i must be unbounded. As a result, { − i || || } { − i || ||} Corollaryp 2.8: αi > 0 for infinitely manyp i. 1 2 √ −αi ||Yi||}iWi Without loss of generality, we assume αi > 0 for all i. Note that lim = W 1 2 ∞ i→∞ ||√ −αi ||Yi||Wi|| ∈ U(Im( T )) Im( T ) and is nonzero. Thus, W = 0. Then we have A ⊂ A A ∞ 6 (Yi) 0 = lim C (7) i→∞ 2 A 1 αi Yi ! − || || 2 p 2 ∗3 = W∞ + lim 1 αi Yi ( Wi) (8) A i→∞ − || || A A q  T ∗3 ∗3 0 2 In consequence, W∞ Im( ) and ( W∞) = lim ( Wi) = since 1 αi Yi i ∈ A A A i→∞ A A { − || ||} T ∗3 is unbounded. Let’s take W := W∞, then W Im( ), ( W ) = 0, and pW = 0. ∈ A A A A 6 2 T T Let Ui := 1 α Yi Wi Im( ), since Wi Im( ). Let δ1 := inf X X − i || ||A ∈ AA ∈ A {||A || | ∈ T Im( )& Xp = 1 , then δ1 > 0 by Propositions 2.4 and 2.5, and δ1 X X for all A || || } || || ≤ ||A || T T 2 X Im( ). Thus, Ui Im( ) i is unbounded since 1 α Yi Wi is unbounded. ∈ A { ∈ AA } { − i || || } Notice that p

∗3 ∗3 Ui + U = (Yi +( Yi) ) (9) A i A A

= (Yi) (10) A · C

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∗3 By the assumption on Yi i, (Yi) i is bounded. Therefore, Ui + U i is bounded. In { } {C } { A i } consequence, ˆ(X) is not proper on Im( T ). C AA This complete the proof of Lemma 2.3. Note that this result is also independently obtained by Tuyen in [48]. In fact, the inverse of (I) is also true, please refer to [48]. Now, we are ready to prove Theorem 1.5 by contradiction. T ∗3 Assume there is some unbounded sequence Ui i Im( ) such that Ui + U i is { } ⊂ AA { A i } bounded. Similar to before, there is some positive real number β > 0 such that

∗3 2 β Ui + U (11) ≥ || A i || 2 T ∗3 ∗3 2 = Ui +2U U + U (12) || || i A i ||A i || T ∗3 2 2 (Ui Ui ) Ui A (13) ≥ || || − U ∗3 2 ||A i || Herein, U ∗3 > 0 for almost all i since β < . In consequence, we have ||A i || ∞ T ∗3 Ui Ui lim A ∗3 = 1 (14) i→∞ Ui U − || ||· ||A i || Ui Ui As i U, we can assume it has a limit. Let U∞ := lim U. It is evident that ||Ui|| i→∞ ||Ui|| { } ⊂ ∈ ∗3 T T AUi Im( ) is a closed set. So U∞ Im( ). Similarly, we consider a limit of ∗3 i. ||AUi || AA ∈ AA ∗3 ∗3 { } AU∞ AUi Without loss of generality, we can assume U ∗3 = lim ∗3 . Then we have ||A ∞ || i→∞ ||AUi || U T U ∗3 U T U ∗3 ∞ ∞ i i (15) A∗3 = lim A ∗3 = 1 U i→∞ Ui U − ||A ∞ || || || ·||A i || Thus, there is some number γ > 0 such that

U + γ U ∗3 = 0 (16) ∞ A ∞ By U Im( T ), we take a Z Im( T ) such that U = Z . By equation (16) ∞ ∈ AA ∞ ∈ A ∞ A ∞ 0 = U + γ U ∗3 (17) ∞ A ∞ = ( + γ (U )∆2)U (18) I A ∞ ∞ = ( + γ ( Z )∆2) Z (19) I A A ∞ A ∞ This is impossible, since ( +γ ( Z )∆2) is invertible by Proposition 2.2 and U = Z U I A A ∞ ∞ A ∞ ∈ is not zero. Therefore, for a D-map (X) := X +( X)∗3, the map ˆ(X) := X + X∗3 must D A D A be proper on Im( T ). In consequence, each D-map must be proper. AA

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III. DISCUSSIONAND RELATED WORK

Recall the history of , there are many excellent works. First of all, Keller proposed J C J C for Kn = C2 in 1939 [35]. And Abhyankar gave its modern style in his lectures [2], [3]. Let ( )k denote (K, n) in which the polynomial degrees are not greater than k. A remarkable J C J C progress is Wang’s result [51] that ( )2 is true for arbitrary field of characteristic = 2. This J C 6 led to the studies on the reduction of to specific ( )k for some small integer k. In the J C J C line of this, Yagˇzev [55] and independently Bass et al. [7] showed that ( )3 implies . J C J C Furthermore, Dru˙zkowski [21] showed that it suffices to show D-maps being automorphism for . Anyway, it cannot make a reduction of to ( )2 [13] on the field C. That is, we J C J C J C must solve ( )3 to the end of . Anyway, many studies [24], [30] showed that it suffices J C J C to prove specific structure D-maps for . Furthermore, also appeared to be connected to J C J C questions in noncommutative algebra, for example, is equivalent to the Dixmier Conjecture J C which asserts that each endomorphism of the Weyl algebra is surjective (hence an automorphism) [5], [9], [50]. is proved equivalent to various conjectures, such as, Kernel Conjecture [29], J C Hessian Conjecture [30], [37], Eulerian Conjecture [4], etc. There are many partial results of J C on special categories of polynomials, for instance, the “non-negative coefficients” D-map [23], D-maps in low dimension space [10], [16], a special class of D-maps in dimension 9 [56], tame automorphisms [19], [46], [47], etc, refer to surveys [7], [20], [28], [29], [54] for more related results. There were some studies about for fixed number of variables, even for 2-variables, J C such as, [38] for 2-variables 100-degree, sufficient conditions via polynomial flows in [8] J C ≤ for (R, 2), a Hamiltonian flows approach in [14] for (C, 2). J C J C must depend on Jacobian Condition, polynomial type, and the number field of charac- J C teristic 0. For the Keller maps over number fields of characteristic 0, injectivity always implies subjectivity [6], [7], [36], [52], [55]. But, this property completely fails for the nonpolynomial

x1 −x1 maps, already for n = 2. There is a counterexample in [7]: F1(X) = e , F2 = x2e whose 2 Jacobian is 1, but (C ) excludes exactly the axis x1 = 0. That is, injectivity does not mean F surjectivity for a generic analytic map. Even for rational maps, there are counterexamples in [22]. As to the zero characteristic condition, there are counterexamples in [3], [7] for of J C characteristic > 0. The Jacobian Condition also cannot be relaxed. A generalization of J C is the real Jacobian problem [42] (also called strong Jacobian Conjecture in [41]), that is, whether a polynomial mapping : R2 R2 with a nonvanishing Jacobian determinant is an F 7→

May 19, 2020 DRAFT 9 automorphism. The strong Jacobian Conjecture has a negative answer [41]. Pinchuk presented 2 a beautiful example of a non-injective polynomial mapping (x1, x2) of R into itself, of F 2 degree (x1, x2) = (10, 40), whose Jacobian determinant is everywhere positive on R . Therefore, Polynomial Automorphism Theorem (PAT for short) is the best in all of what we can get. From PAT, it immediately gets that the Roll Theorem is true for polynomial functions over any algebraically closed field. Let K be an algebraically closed field of characteristic zero, n n K[X] . Then the determinant det(J (X)) is a polynomial and must have an X0 K such P ∈ P ∈ that det(JP (X0)) = 0 if det(JP (X)) is not a constant. By Theorem 1.6, using contradiction argument we can obtain High Dimension Roll Theorem 3.1: If there are X = Y Kn such that (X) = (Y ) then 6 ∈ P P there is some X0 such that det(JP (X0))=0. In this theorem, the requirement “algebraically closed” is necessary. Otherwise, the Pinchuk’s counterexample will be a counterexample over R2. Another immediate result of PAT is that the inverse flows are actually polynomials in X and t. So, high order Lie derivatives vanish at some stages. That is, the Lie derivatives are locally nilpotent or finite [26], [27]. Therefore, it actually gives a termination criterion for computing inverse polynomial through Lie derivatives [40]. When a polynomial map is an automorphism, there are several different approaches to the inversion formulas. An early one is the Abhyankar-Gurjar inversion formula [3]. In [7] Bass et al. presented a formal expansion for the inverse. Nousiainen and Sweedler [40] provided an inversion formula through Lie derivatives. For specific polynomials, Wright [53] and respectively Zhao [57], [58] gave advanced inversion formals. Anyway, the degree of inverse polynomial is bounded by deg(F −1) deg(F )n−1 [7], [31], [44]. ≤ Besides Jacobian Condition, van den Essen [25] using Gr¨obner base gave an algebraic criterion for the invertibility of polynomial maps.

Essen Theorem 3.2 ( [25]): Any map F = (F1(X),...,Fn(X)) K[X] on arbitrary field ∈ K is an automorphism iff there are polynomials G1(Y ),... , Gn(Y ) K[Y ] such that Y1 ∈ − F1(X),...,Yn Fn(X) and X1 G1(Y ),...,Xn Gn(Y ) generate the same polynomial idea − − − in K[X,Y ]. This criterion is not limited to the characteristic zero cases but holds in all characteristics. At the same time Theorem 3.2 also provides an algorithm to decide if a polynomial map has an inverse and compute the inverse if it exists. The theory of Gr¨obner bases for polynomial ideals

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[17] is the foundation of the Essen Theorem. In contrast with this, PAT is an analytic criterion for the global invertibility of polynomial maps. In particular, PAT can be efficiently implemented for sparse polynomial maps, by testing the Jacobian Condition through random inputs.

IV. CONCLUDING REMARKS

In this study, we gave an affirmative answer to Jacobian Conjecture. Based on D-map, our proof used algebraic property like nilpotency property. To study the direction changing tendency of the unbounded sequence Yi i, we used an optimization method to obtain the key limit equations { } (6), (15) and (16). Jacobian Conjecture is an problem. It is no surprise to use algebraic methods in the proof, but the optimization method is really an extra auxiliary. So, this proof is said as an optimization-based method. Our proof takes much advantage of D-maps’ nice algebraic properties. Although Yagˇzev map is a little extension of D-map, it has no such good properties. At least, so far our proof does not work on Yagˇzev map for which we cannot get a necessary condition to nonproperness like Lemma 2.3. From the proof, we can clearly see how and why Jacobian Condition makes D-map being proper and so injective, like the proof for quadratic polynomial maps. However, we should note that an injective polynomial map in R[X]n is not necessarily an automorphism. The automorphism property of D-maps in R[X]n is not a direct result of their injectivity but derived from a series of deductions involved with Lefschetz Principle, Dru˙zkowski’s reduction, automorphism property of D-maps in C[X]n, etc. n Given a polynomial map (X) := ( 1(X),..., n(X)) K[X] , (X) is an automorphism P P P ∈ P n n n n from K onto K iff the induced endomorphism : K[X] K[X] by (Xi)= i(X) for RP 7→ RP P i =1,...,n is an automorphism of the ring K[X]n. If we consider the derivatives as algebraic operators, then the Jacobian Conjecture is purely an algebraic problem. So a purely algebraic proof is really an interesting thing. It is already known that an analytic map satisfying Jacobian Condition is not necessarily a diffeomorphism. In fact, even for the rational maps, the Jacobian Condition is not a sufficient condition for this type of map being diffeomorphisms. For analytic maps over Euclidean space, Hadamard’s Diffeomorphism Theorem has provided a nice criterion for diffeomorphism. However, this is not a computable approach like Jacobian Condition. To the best of our knowledge, so far there is no computable method to directly check the properness of a map. In practice, we may need to computably determine whether a given concrete map is a diffeomorphism. In such context, Hadamard’s Diffeomorphism Theorem helps a little and the Jacobian Condition is not a correct criterion for nonpolymomial maps. In the physical world,

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we are usually concerned with elementary maps which are composed of elementary expressions like eX ,Xp, sin X, cos X, etc. So for analytic maps, it is natural to ask whether there is some computable diffeomorphism criterion. Another basic question about polynomial automorphisms is how many of them, or what is the ratio of polynomial automorphisms to all polynomials, or what is their distribution. By Weierstrass Approximation Theorem, each continuous real function on some closed interval can be uniformly approximated by polynomials. In comparison with this, it is an interesting problem whether the automorphism polynomials are dense in the set of all analytic diffeomorphisms. The study of the injectivity has given rise to several surprising results and interesting relations in various directions and from different perspectives. In this study, we proved Jacobian Conjecture by showing the injectivity of D-maps. In fact, the injectivity itself has received attention from not only the mathematical field but also by the economic field [45], physical field [1], and chemical field [39]. This study only verifies the injectivity of a special class of polynomial maps in R[X]n. The proof heavily depends on the Jacobian Condition and the form of D-map. It has been proved that the Samuelson Conjecture in [45] is true for any polynomial map [18] and arbitrary rational map [12], but fails for generic analytic maps [32]. For a general differentiable map F on Rn, Chamberland Conjecture [15] asserts if all the eigenvalues of JF(X) are bounded away from zero then F must be injective. This conjecture is still open.

V. ACKNOWLEDGEMENT

The author would like to thank Tuyen Trung Truong for his counterexample [49] to a previous aproach. The author also would like to thank Chengling Fang for her help to find out the exact error in the previously invalid proof. This work was partially supported by NSF of China (No. 61672488), CAS Youth Innovation Promotion Association (No. 2015315), National Key R&D Program of China (No. 2018YFC0116704).

REFERENCES

[1] A.ABDESSELAM, The Jacobian conjecture as a problem of perturbative quantum field theory, Annales Henri Poincar 4 (2003), 199–215.

[2] S.S.ABHYANKAR, Lectures on expansion techniques in algebraic geometry. with notes by balwant singh, Tata Institute of Fundamental Research Bombay (1977).

[3] S.S.ABHYANKAR, Lectures in algebraic geometry, Notes by Chris Christensen,Purdue University (1974).

[4] K.ADJAMAGBO and A. VAN DEN ESSEN, Eulerian systems of partial differential equations and the Jacobian conjecture, Journal of Pure and Applied Algebra 74 (1991), 1–15.

May 19, 2020 DRAFT 12

[5] P. K. ADJAMAGBO and A. V. D. ESSEN, A proof of the equivalence of the Dixmier, Jacobian and Poisson conjectures, Acta Mathematica Vietnamica 32 (2007), 205–14.

[6] A. BAILYNICKI-BIRULA and M. ROSENLICHT, Injective morphisms of real algebraic varieties, Proceedings of the American Mathematical Society 13 (1962), 200–203.

[7] H.BASS, E.H.CONNELL, and D.WRIGHT, The Jacobian conjecture: Reduction of degree and formal expansion of the inverse, Bulletin of the American Mathematical Society 7 (1982), 287–330.

[8] H.BASS and G. H. MEISTERS, Polynomial flows in the plane, Advances in 55 (1985), 173–208.

[9] A.BELOVKANEL and M. KONTSEVICH, The Jacobian conjecture is stably equivalent to the Dixmier conjecture, Moscow Mathematical Journal 7 (2007), 209–218.

[10] M. D. BONDT and A. VAN DEN ESSEN, The Jacobian conjecture: Linear triangularization for homogeneous polynomial maps in dimension three, Journal of Algebra 294 (2005), 294–306.

[11] S.BOYD and L. VANDENBERGHE, Convex Optimization, Cambridge University Press, 2004.

[12] L.A.CAMPBELL, Rational Samuelson maps are univalent, Journal of Pure and Applied Algebra 92 (1994), 227–240.

[13] L.A.CAMPBELL, A condition for a polynomial map to be invertible, Mathematische Annalen 205 (1973), 243–248.

[14] L.A.CAMPBELL, Jacobian pairs and Hamiltonian flows, Journal of Pure and Applied Algebra 115 (1997), 15–26.

[15] M.CHAMBERLAND and G. MEISTERS, A mountain pass to the Jacobian conjecture., Canadian Mathematical Bulletin 41 (1998), 442–451.

[16] C.C.CHENG, Power linear Keller maps of dimension four, Journal of Pure and Applied Algebra 169 (2002), 153–158.

[17] D.A.COX,J.LITTLE, and D. O’SHEA, Ideals, Varieties, and Algorithms, Springer, 1997.

[18] A. V. DEN ESSEN and T. PARTHASARATHY, Polynomial maps and a conjecture of Samuelson, Linear Algebra and its Applications 177 (1992), 191–195.

[19] V. DRENSKY, A. VAN DEN ESSEN, and D.STEFANOV, New stably tame automorphisms of polynomial algebras, Journal of Algebra 226 (2000), 629–638.

[20] L. M. DRUZKOWSKI˙ , Partial results and equivalent formulations of the Jacobian conjecture, Rendiconti Del Seminario Matematico (1997), 275–282.

[21] L.M.DRUZKOWSKI˙ , An effective approach to Keller’s Jacobian conjecture, Mathematische Annalen 264 (1983), 303–313.

[22] L.M.DRUZKOWSKI˙ , The Jacobian conjecture: survey of some results, Banach Center Publications 31 (1995), 163–171.

[23] L. M. DRUZKOWSKI˙ , The Jacobian conjecture in case of “non-negative coefficients”, Annales Polonici Mathematici 66 (1997), 67–75.

[24] L. M. DRUZKOWSKI˙ , The Jacobian conjecture: symmetric reduction and solution in the symmetric cubic linear case, Annales Polonici Mathematici 87 (2005), 83–92.

[25] A. VAN DEN ESSEN, A criterion to decide if a polynomial map is invertible and to compute the inverse, Communications in Algebra 18 (1990), 3183–3186.

[26] A. VAN DEN ESSEN, Locally finite and locally nilpotent derivations with applications to polynomial flows and polynomial morphisms, Proceedings of the American Mathematical Society 116 (1992), 861–871.

[27] A. VAN DEN ESSEN, Locally finite and locally nilpotent derivations with applications to polynomial flows, morphisms and

Ya-actions. II., Proceedings of the American Mathematical Society 121 (1994), 667–678.

[28] A. VAN DEN ESSEN, Polynomial automorphisms and the Jacobian conjecture, in Progress in Mathematics Vol 190, Boston; Berlin: Birkh¨auser, 2000.

[29] A. VAN DEN ESSEN, Polynomial automorphisms, and the Jacobian conjecture, The Mathematical Gazette 85 (2001), 572.

[30] A. VAN DEN ESSEN, A reduction of the Jacobian conjecture to the symmetric case, Proceedings of the American Mathematical Society 133 (2005), 2201–2205.

May 19, 2020 DRAFT 13

[31] J. FURTER, On the degree of the inverse of an automorphism of the affine space, Journal of Pure and Applied Algebra 130 (1998), 277–292.

[32] D.GALE and H. NIKAIDO, The Jacobian matrix and global univalence of mapping, Mathematische Annalen 159 (1965), 81–93.

[33] W. B. GORDON, On the diffeomorphisms of Euclidean space, American Mathematical Monthly 79 (1972), 755–759.

[34] J.HADAMARD, Sur les transformations ponctuelles, Bull.soc.math.france (1906), 71–84.

[35] O.KELLER, Ganze Cremona-transformationen, Monatshefte f¨ur Mathematik 47 (1939), 299–306.

[36] K. KURDYKA and K. RUSEK, Surjectivity of certain injective semialgebraic transformations of Rn, Mathematische Zeitschrift 200 (1988), 141–148.

[37] G. MENG, Legendre transform, Hessian conjecture and tree formula, Applied Mathematics Letters 19 (2006), 503–510.

[38] T. T. MOH, On the Jacobian conjecture and the configurations of roots, Journal F¨ur Die Reine Und Angewandte Mathematik 1983 (1983), 140–213.

[39] S. MULLER,J.HOFBAUER,and G.REGENSBURGER, On the bijectivity of families of exponential/generalized polynomial maps, SIAM Journal on Applied Algebra and Geometry 3 (2019), 412–438.

[40] P. NOUSIAINEN and M. SWEEDLER, Automorphisms of polynomial and power series rings, Journal of Pure and Applied Algebra 29 (1983), 93–97.

[41] S.PINCHUK, A counterexamle to the strong real Jacobian conjecture, Mathematische Zeitschrift 217 (1994), 1–4.

[42] J.D.RANDALL, The real Jacobian problem, 1983, pp. 411–414.

[43] W. RUDIN, Injective polynomial maps are automorphisms, American Mathematical Monthly 102 (1995), 540–543. n [44] K.RUSEK and T. WINIARSKI, Polynomial automorphisms of C , 661 (1984).

[45] P. A. SAMUELSON, Prices of factors and good in general equilibrium, Review of Economic Studies 21 (1953), 1–20.

[46] I.SHESTAKOV and U. UMIRBAEV, The tame and the wild automorphisms of polynomial rings in three variables, Journal of the American Mathematical Society 17 (2004), 197–227.

[47] M. K.SMITH, Stably tame automorphisms, Journal of Pure and Applied Algebra 58 (1989), 209–212.

[48] T. T. TRUONG, Some observations on the properness of identity plus linear powers, arXiv:2004.03309, 2020.

[49] T. T. TRUONG, Some observations on the properness of identity plus linear powers: part 2, arXiv:2005.02260, 2020.

[50] Y. TSUCHIMOTO, Endomorphisms of Weyl algebra and p-curvatures, Osaka Journal of Mathematics 42 (2005), 435–452.

[51] S.S.WANG, A Jacobian criterion for separability, Journal of Algebra 65 (1980), 453–494.

[52] T. WINIARSKI, Inverse of polynomial automorphisms of Cn, Journal of the Institute of Image Information & Television Engineers 27 (1979), 9–11.

[53] D. WRIGHT, Formal inverse expansion and the Jacobian conjecture, Journal of Pure and Applied Algebra 48 (1987), 199–219.

[54] D.WRIGHT, The Jacobian conjecture: ideal membership questions and recent advances, (2005), 261–276.

[55] A. V. YAGZEVˇ , Keller’s problem, Siberian Mathematical Journal 21 (1980), 747–754.

[56] D.YAN, A note on the Jacobian conjecture, Linear Algebra and its Applications 435 (2011), 2110–2113.

[57] W. ZHAO, Recurrent inversion formulas, in Some Properties and Open Problems of Hessian Nilpotent polynomials, In preparation. Department of Mathematics, Illinois State University, Normal, IL 61790-4520. E-mail: [email protected].

[58] W. ZHAO, Inversion problem, Legendre transform and inviscid Burgers’ equations, Journal of Pure and Applied Algebra 199 (2005), 299–317.

May 19, 2020 DRAFT