On Harmonic Conjugates in Full Algebraic Combinatorial Geometries

AbstractThe points of a dense algebraic combinatorial geometry are equivalence classes of transcendentals over a field F in the algebraic closure of a transcendental extension of F. Two transcendentals represent the same point when they are algebraically dependent over F.If x and y are two algebraically independent transcendentals over F the points of the algebraic closure of the field F(x, y) belong to a line. Planes are defined similarly.By analogy with classical projective geometry, we define harmonic conjugates with respect to 2 points on a line. We prove the existence and uniqueness of the harmonic conjugate of a point with respect to two other points on a line.The main tool is a lemma by Ingleton and Main in [3]