Polynomials Associated with Newton Polygons

The polynomial used to define a polygon can be recovered, but more usefully so can those restrictions of that polynomial to parts of the polygon, the so-called face functions in particular.

Note that most of these functions will return an error if N was not defined in terms of a polynomial.

HasPolynomial(N) : NwtnPgon -> BoolElt

True if and only if N was defined as the Newton polygon of some polynomial.

Polynomial(N) : NwtnPgon -> RngElt

The polynomial used to define N.

ParentRing(N) : NwtnPgon -> Rng

The parent ring of the polynomial of N.

FaceFunction(F) : NwtnPgon,Tup -> RngElt

If N is defined by a polynomial in two variables f this returns those monomial terms of f whose corresponding Newton points lie on the face F. On the other hand, if N is determined by a univariate polynomial over a series ring, this returns the univariate polynomial supported on the face F.

[Future release] IsDegenerate(F) : NwtnPgon,Tup -> BoolElt

False if and only if the face function of N along F is squarefree.

[Future release] IsDegenerate(N) : NwtnPgon -> BoolElt

True if and only if N is degenerate along some face.