Extending the classical result that the roots of a polynomial with coefficients in $\mathbf{ C}$ are continuous functions of the coefficients of the polynomial, nonstandard analysis is used to prove that if $\mathcal{ F} = \{ f_{ \lambda} :\lambda \in \Lambda\}$ is a set of polynomials in $\mathbf{ C}[t_1,\ldots, t_n]$ and if $^*\mathcal{ G} = \{ g_{ \lambda} :\lambda \in \Lambda\}$ is a set of polynomials in $^*\mathbf{ C}_0[t_1,\ldots, t_n]$ such that $g_{ \lambda}$ is an infinitesimal deformation of $f_{ \lambda}$ for all $\lambda \in \Lambda$, then the nonstandard affine variety $^*V_0(\mathcal{ G})$ is an infinitesimal deformation of the affine variety $V(\mathcal{ F})$.
Categories: math.AG math.LO