arXiv subjects

Cynthia Vinzant

Publications and source records attributed to Cynthia Vinzant.

39 records · Page 3Linked to original sources

arXiv2011

Real radical initial ideals

We explore the consequences of an ideal I of real polynomials having a real radical initial ideal, both for the geometry of the real variety of I and as an application to sums of squares representations of polynomials. We show that if in_w(I) is real radical for a vector w in the tropical variety, then w is in the logarithmic set of the real variety. We also give algebraic sufficient conditions for w to be in the logarithmic limit set of a more general semialgebraic set. If in addition the entries of w are positive, then the corresponding quadratic module is stable. In particular, if in_w(I) is real radical for some positive vector w then the set of sums of squares modulo I is stable. This provides a method for checking the conditions for stability given by Powers and Scheiderer.

math.AG↗

arXiv2011

Lower Bounds for Optimal Alignments of Binary Sequences

In parametric sequence alignment, optimal alignments of two sequences are computed as a function of the penalties for mismatches and spaces, producing many different optimal alignments. Here we give a 3/(2^{7/3}π^{2/3})n^{2/3} +O(n^{1/3} \log n) lower bound on the maximum number of distinct optimal alignment summaries of length-n binary sequences. This shows that the upper bound given by Gusfield et. al. is tight over all alphabets, thereby disproving the "square root of n conjecture". Thus the maximum number of distinct optimal alignment summaries (i.e. vertices of the alignment polytope) over all pairs of length-n sequences is Theta(n^{2/3}).

q-bio.GN↗

arXiv2011

Quartic Curves and Their Bitangents

A smooth quartic curve in the complex projective plane has 36 inequivalent representations as a symmetric determinant of linear forms and 63 representations as a sum of three squares. These correspond to Cayley octads and Steiner complexes respectively. We present exact algorithms for computing these objects from the 28 bitangents. This expresses Vinnikov quartics as spectrahedra and positive quartics as Gram matrices. We explore the geometry of Gram spectrahedra and we find equations for the variety of Cayley octads. Interwoven is an exposition of much of the 19th century theory of plane quartics.

math.AG↗