113 research outputs found
Advances on the Bessis-Moussa-Villani Trace Conjecture
A long-standing conjecture asserts that the polynomial has nonnegative coefficients whenever is a positive
integer and and are any two positive semidefinite
Hermitian matrices. The conjecture arises from a question raised by Bessis,
Moussa, and Villani (1975) in connection with a problem in theoretical physics.
Their conjecture, as shown recently by Lieb and Seiringer, is equivalent to the
trace positivity statement above. In this paper, we derive a fundamental set of
equations satisfied by and that minimize or maximize a coefficient of
. Applied to the Bessis-Moussa-Villani (BMV) conjecture, these equations
provide several reductions. In particular, we prove that it is enough to show
that (1) it is true for infinitely many , (2) a nonzero (matrix) coefficient
of always has at least one positive eigenvalue, or (3) the result
holds for singular positive semidefinite matrices. Moreover, we prove that if
the conjecture is false for some , then it is false for all larger .Comment: 12 page
Finite Groebner bases in infinite dimensional polynomial rings and applications
We introduce the theory of monoidal Groebner bases, a concept which
generalizes the familiar notion in a polynomial ring and allows for a
description of Groebner bases of ideals that are stable under the action of a
monoid. The main motivation for developing this theory is to prove finiteness
theorems in commutative algebra and its applications. A major result of this
type is that ideals in infinitely many indeterminates stable under the action
of the symmetric group are finitely generated up to symmetry. We use this
machinery to give new proofs of some classical finiteness theorems in algebraic
statistics as well as a proof of the independent set conjecture of Hosten and
the second author.Comment: 24 pages, adds references to work of Cohen, adds more details in
Section
Algebraic Characterization of Uniquely Vertex Colorable Graphs
The study of graph vertex colorability from an algebraic perspective has
introduced novel techniques and algorithms into the field. For instance, it is
known that -colorability of a graph is equivalent to the condition for a certain ideal I_{G,k} \subseteq \k[x_1, ..., x_n]. In this
paper, we extend this result by proving a general decomposition theorem for
. This theorem allows us to give an algebraic characterization of
uniquely -colorable graphs. Our results also give algorithms for testing
unique colorability. As an application, we verify a counterexample to a
conjecture of Xu concerning uniquely 3-colorable graphs without triangles.Comment: 15 pages, 2 figures, print version, to appear J. Comb. Th. Ser.
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