Abstract. Let k be an algebraically closed field of characteristic zero. Let H : k2 →
1. H: k^2 -> k^2이 Jac(H) = c (const != 0)를 만족하는 다항식 매핑이고,
일직선 l 이 있고, l에서 H|l : l -> k^2가 주어지면, H는 다항식 아벨로미즘을 만족한다는 것이다.
H(x,y) = (f(x,y), g(x,y))라고 하자.
그런데 Jac(H)(x,y) = ∂f/∂x * ∂g/∂y - ∂f/∂y * ∂g/∂x 이므로,
일직선 l에서 인젝티브인 H|x, y=0 = (f(x, 0), g(x, 0))이 성립해야한다.
여기서, H(x,y) = (H1(x,y), G(x,y))라고 하자.
그런데 Abhyankar-Moh theorem에 따르면 G| x, y=0 = (x, 0) 이므로 G는 주어진 조건을 만족한다.
따라서, H는 다항식 아벨로미즘이다.
Abstract. Let k be an algebraically closed field of characteristic zero. Let H : k2 →
arXiv:alg-geom/9305008v1 19 May 1993INJECTIVITY ON ONE LINEby Janusz Gwo´zdziewiczAbstract. Let k be an algebraically closed field of characteristic zero.
Let H : k2 →k2 be a polynomial mapping such that the Jacobian Jac H is a non-zero constant. Inthis note we prove, that if there is a line l ⊂k2 such that H|l : l →k2 is an injection,then H is a polynomial automorphism.1.
Main resultLet k be an algebraically closed field of characteristic zero. Put k∗= k \ {0}.By Aut k2 we denote the group of polynomial automorphisms i.e.
all mappingsH = (f, g) : k2 →k2, f, g ∈k[x, y] for which there exists an inverse polynomialmapping. Remind that for H ∈Aut k2 the Jacobian Jac H is a non-zero constant.The famous Keller conjecture states that any polynomial mapping which has thenon-zero constant Jacobian is a polynomial automorphism.Theorem 1.1.
Let H : k2 →k2 be a polynomial mapping such that JacH ∈k∗. Ifthere exists a line l ⊂k2 such that H|l : l →k2 is injective then H is a polynomialautomorphism.The proof of the above theorem is given in Section 2 of this note.
Let us notehere that a weaker result (injectivity on three lines) has been proved recently in [4].2. Proof of the theoremThe proof of our result is based on the Abhyankar–Moh theorem and on someproperties of Newton’s polygon which we quote below.If f is a polynomial in k[x, y] then Sf denotes the support of f, that is, Sf isthe set of integer points (i, j) such that the monomial xiyj appears in f with anon-zero coefficient.
We denote by Nf the convex hull (in the real space R2) ofSf ∪{(0, 0)}. The set Nf is called (see [2]) Newton’s polygon of f.Theorem 2.1.
( [2], [3, theorem 3.4.]) Let f, g be polynomials in k[x, y].
Assumethat Jac(f, g) is a non-zero constant and deg f > 1, deg g > 1. Then the polygonsNf and Ng are similar, that is Ng = deg gdeg f Nf.The following lemma is easy to check, so we omit the proof.Lemma 2.2.
Let f, g be polynomials in k[x, y] with Jac(f, g) ∈k∗. If deg f ≤1or deg g ≤1 then the mapping (f, g) is a polynomial automorphism.Typeset by AMS-TEX1
2BY JANUSZ GWO´ZDZIEWICZLemma 2.3. Let H = (f, g) be a polynomial mapping such that Jac H is a non-zeroconstant.
If H(x, 0) = (x, 0) then H ∈Aut k2.Proof. First suppose that deg f > 1, deg g > 1.
We havef(x, 0) = x(1)g(x, 0) = 0. (2)From (1) the point (1, 0) belongs to the polygon Nf, so by theorem 2.1 ( deg gdeg f , 0) ∈Ng.
This means that the polynomial g contains some monomials of the form xi,i > 0 with non-zero coefficients but this is a cotradiction with (2).We have deg f ≤1 or deg g ≤1 and by lemma 2.2 H is a polynomial automor-phism.Proof of theorem 1.1. Without loss of generality we may assume that the line lhas an equation y = 0.
Otherwiese we replace H by H ◦L where L is an affineautomorphism such that L({y = 0}) = l.Put γ(x) = H(x, 0). By assumptions of the theorem the mapping γ : k →k2 isan injection and γ′(x) ̸= 0 for x ∈k, hence γ is an embedding of the line in theplane.
By Abhyankar-Moh theorem [1] there exist an automorphism H1 ∈Aut k2such that γ(x) = H1(x, 0). Let G = H1 ◦H.
We get Jac G = Jac H−11Jac H is anon-zero constant and G(x, 0) = (x, 0), so by lemma 2.3 G ∈Aut k2. ThereforeH = H1 ◦G is a polynomial automorphism.References1.
S. S. Abhyankar, T. T. Moh, Embeddings of the line in the plane, Journal Reine Angew. Math.276 (1975), 149–166.2.
S. S. Abhyankar, Expansion Techniques in Algebraic Geometry, Tata Inst. of FundamentalResearch, Bombay, 1977.3.
J. Lang, Newton polygons of jacobian pairs, Journal of Pure and Applied Algebra 72 (1991),39–51.4. A. van den Essen, H. Tutaj, A remark on the two-dimensional Jacobian conjecture, (preprint,October, 1992).Injektywno´s´c na jednej prostejStreszczenie.Niech k b ιedzie cia lem algebraicznie domkni ιetym charakterystyki zero.
Niech H : k2 →k2b ιedzie odwzorowaniem wielomianowym kt´orego Jakobian Jac H jest sta l ιa r´o˙zn ιa od zera.Wpracy dowodzimy, ˙ze je´sli istnieje prosta l ⊂k2 na kt´orej H|l : l →k2 jest injekcj ιa, to H jestautomorfizmem wielomianowym.Department of Mathematics,Technical University,Al. 1000 LPP 7, 25–314 Kielce,Poland
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