A nonstandard proof of continuity of affine varieties

Abstract: Extending the classical result that the roots of a polynomial with coefficients in

m a t h b f C

are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if

m a t h c a l F = f_{l a m b d a} : l a m b d a i n L a m b d a

is a set of polynomials in

m a t h b f C [t_{1}, l d o t s, t_{n}]

and if

* m a t h c a l G = g_{l a m b d a} : l a m b d a i n L a m b d a

is a set of polynomials in

* m a t h b f C_{0} [t_{1}, l d o t s, t_{n}]

such that

g_{l a m b d a}

is an infinitesimal deformation of

f_{l a m b d a}

for all

l a m b d a i n L a m b d a

, then the nonstandard affine variety

* V_{0} (m a t h c a l G)

is an infinitesimal deformation of the affine variety

V (m a t h c a l F)

.

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