Newton Polytopes of Nondegenerate Quadratic Forms

Applications of quadratic D-forms to generalized quadratic forms

In this paper, we study generalized quadratic forms over a division algebra with involution of the first kind in characteristic two. For this, we associate to every generalized quadratic from a quadratic form on its underlying vector space. It is shown that this form determines the isotropy behavior and the isometry class of generalized quadratic forms.

Newton Polytopes and Witness Sets

We present two algorithms that compute the Newton polytope of a polynomial f defining a hypersurface H in Cn using numerical computation. The first algorithm assumes that we may only compute values of f—this may occur if f is given as a straightline program, as a determinant, or as an oracle. The second algorithm assumes that H is represented numerically via a witness set. That is, it computes ...

Elimination Theory and Newton Polytopes

Pruning Algorithms for Pretropisms of Newton Polytopes

Pretropisms are candidates for the leading exponents of Puiseux series that represent positive dimensional solution sets of polynomial systems. We propose a new algorithm to both horizontally and vertically prune the tree of edges of a tuple of Newton polytopes. We provide experimental results with our preliminary implementation in Sage that demonstrates that our algorithm compares favorably to...

New Normal Forms for Levi-nondegenerate Hypersurfaces

In this paper we construct a large class of new normal forms for Levi-nondegenerate real hypersurfaces in complex spaces. We adopt a general approach illustrating why these normal forms are natural and which role is played by the celebrated Chern-Moser normal form [CM74]. The latter appears in our class as the one with the ”maximum normalization” in the lowest degree. However, there are other n...