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In this extended abstract, we study the properties of ultraquadrics associated with automorphisms of the field K(\alpha )(t1, . . . , tn), defined by linear rational (with common denominator)\udor by polynomial (with polynomial inverse) coordinates. We conclude that ultraquadrics related to\udpolynomial automorphisms can be characterized as varieties K−isomorphic to linear varieties, while\udultraquadrics arising from projective automorphisms are isomorphic to the Segre embedding of a\udblowup of the projective space along an ideal and, in some general case, linearly isomorphic to a toric\udvariety. This information helps us to compute a parametrization of some ultraquadrics

Ultraquadrics associated to affine and projective automorphisms