A Taylor variety is the set of fixed order Taylor polynomials of rational functions of a given degree and number of variables. Deciding whether a polynomial belongs to this variety is related to the Padé approximation problem, a central question in computer algebra. I will discuss some results concerning dimension and defining ideals of Taylor varieties. In one variable, Taylor varieties are classical algebraic varieties. I will discuss the case of multivariate Taylor varieties, in particular hypersurfaces, some of which have vanishing Hessian. In three and more variables, some Taylor varieties are defective (their dimension is smaller than expected).
