In this article, we present the theory of standard bases for modules over polynomial subalgebras based on arbitrary gradings (induced by an additive monoid Gamma) which we called S Gamma-bases. This paper deals with the problem of construction of S Gamma-bases for submodules of free module over polynomial subalgebras. For this purpose, we present an algorithm based on S Gamma-normal form and discuss it for some special gradings.

The process of replacing indeterminates in a Polynomial with other polynomials is the polynomial composition. Homogeneous Sagbi bases are the Sagbi bases generated by the subset of homogeneous polynomials. In this article we present adequate and essential criterion on a set of polynomials to guarantee that the composed set \(S\circ \ominus\) is Homogeneous Sagbi basis whenever \(S\) is a Homogeneous Sagbi basis.