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

Return true if and only if the polygon \(N\) was defined as the Newton polygon of some polynomial.

Polynomial(N): NwtnPgon -> RngElt

The polynomial used to define the polygon \(N\).

ParentRing(N): NwtnPgon -> Rng

The parent ring of the polynomial of the polygon \(N\).

IsNewtonPolygonOf(N, f): NwtnPgon, RngElt -> BoolElt

Return whether the newton polygon \(N\) is defined by the polynomial \(f\).

FaceFunction(F): NwtnPgonFace -> RngElt

If the polygon \(N\) 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 \(F\).

IsDegenerate(F): NwtnPgonFace -> BoolElt

Return true if the face function along \(F\) is not squarefree.

IsDegenerate(N): NwtnPgon -> BoolElt

Return true if a face function on some face of \(N\) is degenerate.

IsNondegenerate(F): NwtnPgonFace -> BoolElt

Return false if the face function along \(F\) is not squarefree.

IsNondegenerate(N): NwtnPgon -> BoolElt

Return false if a face function on some face of \(N\) is degenerate.