A new combinatorial proof confirms total positivity of the barycentric subdivision transformation matrix, proves it for interval subdivision, and gives a sufficient condition for TP2 in uniform subdivisions with an application to r-colored barycentric subdivision.
Polynomials, roots, and interlacing
2 Pith papers cite this work. Polarity classification is still indexing.
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abstract
This work is divided into three parts. The first part concerns polynomials in one variable with all real roots. We consider linear transformations that preserve real rootedness, as well as matrices that preserve interlacing. The second part covers polynomials in several variables that generalize polynomials with all real roots. We introduce generating functions and use them to establish properties of a linear transformation. We also consider matrices and matrix polynomials. The third part considers polynomials with complex roots. The two main classes considered are polynomials with all roots in the left half plane (stable polynomials) and those with all roots in the lower half plane (Upper half plane polynomials). These naturally generalize to polynomials in many variables. And, of course, there is much more.
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2026 2verdicts
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Adding a suitable extra point E to P_n(x) produces complete zero interlacing with G_k when the polynomials obey appropriate mixed recurrence relations.
citing papers explorer
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Total positivity of transformation matrices for uniform subdivisions
A new combinatorial proof confirms total positivity of the barycentric subdivision transformation matrix, proves it for interval subdivision, and gives a sufficient condition for TP2 in uniform subdivisions with an application to r-colored barycentric subdivision.
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Separating zeros of polynomials using an added interlacing point
Adding a suitable extra point E to P_n(x) produces complete zero interlacing with G_k when the polynomials obey appropriate mixed recurrence relations.