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作者 NOUSIAINEN, PEKKA SAKARI
書名 ON THE JACOBIAN PROBLEM
說明 49 p
附註 Source: Dissertation Abstracts International, Volume: 43-03, Section: B, page: 0751
Thesis (Ph.D.)--The Pennsylvania State University, 1982
Let C('n) denote the n-dimensional complex affine space. A map (phi):C('n) (--->) C('n) is smooth if its Jacobian has rank n at each point. The Jacobian Conjecture is the claim that a smooth polynomial map (phi):C('n) (--->) C('n) is invertible. (One can show that the inverse, if it exists, is also a polynomial map). Note that locally (phi) does have an inverse, by the analytic inverse function theorem. The Jacobian conjecture is thus a conjectured global inverse function theorem for polynomial maps. Little progress has been made on this conjecture in the more than 40 years since it first appeared in {Ke}. The main general result known is that a smooth proper polynomial map is invertible
The present thesis contains a number of contributions to the Jacobian conjecture. We consider base fields more general than C, and our tools are basic commutative algebra and algebraic geometry. Section 1 introduces the Jacobian problem in the algebraic setting. Sections 2 and 3 are joint work with Moss Sweedler (Cornell University). The invertibility of a smooth polynomial map is shown to be equivalent to "integrability" conditions (local nilpotence and local finiteness) on certain vector fields. Section 4 is a study of the 2-variable polynomial ring C{X,Y}, considered as a Lie algebra where the bracket {f,g} is the determinant of the Jacobian of f and g. The main result extends the local nilpotence result of section 3
Sections 5 and 6 are two attempts to study the Jacobian Conjecture by considering the problem in positive characteristic. Section 5 uses ultraproducts to establish a connection between the possible degrees (generic number of points in fibres) of smooth maps over different base fields. Section 6 gives a characterization of smoothness in characteristic p > 0 in terms of p('th) powers
Section 7 on infinite dimensional algebraic groups is in part motivated by the Jacobian Conjecture. The idea is to consider the "generic" problem, i.e., to consider the infinite dimensional varieties whose points are smooth or invertible endomorphisms of the affine space. We also show that the automorphisms of any finitely generated algebra form an infinite dimensional group. No attempt is made to develop the subject of infinite dimensional algebraic geometry beyond the properties needed to discuss the groups
School code: 0176
Host Item Dissertation Abstracts International 43-03B
主題 Mathematics
0405
Alt Author The Pennsylvania State University
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