3,479 research outputs found
Note on the location of zeros of polynomials
In this note, we provide a wide range of upper bounds for the moduli of the
zeros of a complex polynomial. The obtained bounds complete a series of
previous papers on the location of zeros of polynomials.Comment: 8 page
Twisting Alexander Invariants with Periodic Representations
Twisted Alexander invariants have been defined for any knot and linear
representation of its group. The invariants are generalized for any periodic
representation of the commutator subgroup of the knot group. Properties of the
new twisted invariants are given. Under suitable hypotheses, reciprocality and
bounds on the moduli of zeros are obtained. A topological interpretation of the
Mahler measure of the invariants is presented. Keywords: Knot, twisted
Alexander polynomial, representation shift, Mahler measure.Comment: 21 pages, no figures. Version 2 contains some improvements and
update
Generalization and variations of Pellet's theorem for matrix polynomials
We derive a generalized matrix version of Pellet's theorem, itself based on a
generalized Rouch\'{e} theorem for matrix-valued functions, to generate upper,
lower, and internal bounds on the eigenvalues of matrix polynomials. Variations
of the theorem are suggested to try and overcome situations where Pellet's
theorem cannot be applied.Comment: 20 page
Polynomial cubic differentials and convex polygons in the projective plane
We construct and study a natural homeomorphism between the moduli space of
polynomial cubic differentials of degree d on the complex plane and the space
of projective equivalence classes of oriented convex polygons with d+3
vertices. This map arises from the construction of a complete hyperbolic affine
sphere with prescribed Pick differential, and can be seen as an analogue of the
Labourie-Loftin parameterization of convex RP^2 structures on a compact surface
by the bundle of holomorphic cubic differentials over Teichmuller space.Comment: 64 pages, 5 figures. v3: Minor revisions according to referee report.
v2: Corrections in section 5 and related new material in appendix
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