  We give an explicit weak solution to the Schottky problem, in the spirit of
Riemann and Schottky. For any genus $g$, we write down a collection of
polynomials in genus $g$ theta constants, such that their common zero locus
contains the locus of Jacobians of genus $g$ curves as an irreducible
component. These polynomials arise by applying a specific Schottky-Jung
proportionality to an explicit collection of quartic identities for theta
constants in genus $g-1$, which are suitable linear combinations of Riemann's
quartic relations.
