-----Original Message----- From: math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com [mailto:math-fun-bounces+andy.latto=pobox.com@mailman.xmission.com]On Behalf Of Daniel Asimov Sent: Monday, August 28, 2006 11:48 PM To: math-fun Subject: Re: [math-fun] Possibly naive question about algebraic numbers

Also, given such an integer-polynomial function P: C^n -> C^n such that its Jacobian at each of its roots Z in C^n is non-singular:

QUESTION: What can be said about the size of the (necessarily) 0-dimensional set of such roots?

If you ask the question in projective space, rather than in affine space, you get the answer you want, which is that provided all the roots are simple (that is, the Jacobian is non-singular), then the number of roots is exactly equal to the product of the degrees of the polynomials. If you ask the question in affine space, the best you can say is that the number of roots is <= this product, because you have left out the "roots at infinity". In two dimensions, this goes by the name of Bezout's theorem. Andy Latto andy.latto@pobox.com

(Where of course a root of P means a Z in C^n where P(Z) is the origin in C^n.)

--Dan

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