I How should a projective geometric property be written analytically? A first order formula in the language of fields which expresses a “projective geometric property” is translated, by an algorithm, into a restricted class of formulas in the analytic geometric language of brackets (or invariants). This special form corresponds to statements in synthetic projective geometry and the algorithm is a basic step towards translation back into synthetic geometry.

II How are theorems of analytic geometry proven? Axioms for the theorems of analytic projective geometry are given in the invariant language. Identities derived form Hubert's Nullstellensatz then play a central role in the proof. Prom a proof of an open theorem about “geometric properties”, over all fields, or over ordered fields, an algorithm derives Nullstellensatz identities — giving maximal algebraic simplicity, and maximal information in the proof.

The results support the proposal that computational analytic projective geometry should be carried out directly with identities in the invariant language.