We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.
Published online by Cambridge University Press: 29 December 2009
The Jacobian conjecture was proposed by O.H. Keller in 1939. It asks whether a polynomial endomorphism of ℂn whose Jacobian is constant must be invertible. Despite its simple and reasonable statement, the conjecture has not been proved even in the two dimensional case. In this chapter we show that this conjecture would follow if one could prove that every endomorphism of the Weyl algebra is an automorphism. The chapter opens with a discussion of polynomial maps, which will play a central rôle in the second part of the book. We shall return to the Jacobian conjecture in Ch. 19.
POLYNOMIAL MAPS.
Let F : Kn → Km be a map and p a point of Kn. We say that F is polynomial if there exist F1, …, Fm ∈ K[x1, …, xn] such that F(p) = (F1(p), …, Fm(p)). A polynomial map is called an isomorphism or a polynomial isomorphism if it has an inverse which is also a polynomial map. It is not always the case that a bijective polynomial map has an inverse which is also polynomial. For an example where this does not occur see Exercise 5.1. However, if K = ℂ, every invertible polynomial map has a polynomial inverse. This is proved in [Bass, Connell and Wright; Theorem 2.1].
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.
