34,365 research outputs found
Milnor numbers of projective hypersurfaces and the chromatic polynomial of graphs
The chromatic polynomial of a graph G counts the number of proper colorings
of G. We give an affirmative answer to the conjecture of Read and
Rota-Heron-Welsh that the absolute values of the coefficients of the chromatic
polynomial form a log-concave sequence. We define a sequence of numerical
invariants of projective hypersurfaces analogous to the Milnor number of local
analytic hypersurfaces. Then we give a characterization of correspondences
between projective spaces up to a positive integer multiple which includes the
conjecture on the chromatic polynomial as a special case. As a byproduct of our
approach, we obtain an analogue of Kouchnirenko's theorem relating the Milnor
number with the Newton polytope.Comment: Improved readability. Final version, to appear in J. Amer. Math. So
On Comon's and Strassen's conjectures
Comon's conjecture on the equality of the rank and the symmetric rank of a
symmetric tensor, and Strassen's conjecture on the additivity of the rank of
tensors are two of the most challenging and guiding problems in the area of
tensor decomposition. We survey the main known results on these conjectures,
and, under suitable bounds on the rank, we prove them, building on classical
techniques used in the case of symmetric tensors, for mixed tensors. Finally,
we improve the bound for Comon's conjecture given by flattenings by producing
new equations for secant varieties of Veronese and Segre varieties.Comment: 12 page
Generalized conditional symmetries of evolution equations
We analyze the relationship of generalized conditional symmetries of
evolution equations to the formal compatibility and passivity of systems of
differential equations as well as to systems of vector fields in involution.
Earlier results on the connection between generalized conditional invariance
and generalized reduction of evolution equations are revisited. This leads to a
no-go theorem on determining equations for operators of generalized conditional
symmetry. It is also shown that up to certain equivalences there exists a
one-to-one correspondence between generalized conditional symmetries of an
evolution equation and parametric families of its solutions.Comment: 23 pages, extended versio
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