Abeliants and their application to an elementary construction of Jacobians

The abeliant is a polynomial rule which to each n × n by n + 2 array with entries in a commutative ring with unit associates an n × n matrix with entries in the same ring. The theory of abeliants, first introduced in an earlier paper of the author, is simplified and extended here. Now let J be the Jacobian of a nonsingular projective algebraic curve defined over an algebraically closed field. With the aid of the theory of abeliants we obtain explicit defining equations for J and its group law.