56,452 research outputs found
Quantitative equidistribution for the solutions of systems of sparse polynomial equations
For a system of Laurent polynomials f_1,..., f_n \in C[x_1^{\pm1},...,
x_n^{\pm1}] whose coefficients are not too big with respect to its directional
resultants, we show that the solutions in the algebraic n-th dimensional
complex torus of the system of equations f_1=\dots=f_n=0, are approximately
equidistributed near the unit polycircle. This generalizes to the multivariate
case a classical result due to Erdos and Turan on the distribution of the
arguments of the roots of a univariate polynomial. We apply this result to
bound the number of real roots of a system of Laurent polynomials, and to study
the asymptotic distribution of the roots of systems of Laurent polynomials with
integer coefficients, and of random systems of Laurent polynomials with complex
coefficients.Comment: 29 pages, 2 figures. Revised version, accepted for publication in the
American Journal of Mathematic
Discrepancy of Symmetric Products of Hypergraphs
For a hypergraph , its --fold symmetric
product is . We give
several upper and lower bounds for the -color discrepancy of such products.
In particular, we show that the bound proven for all in [B. Doerr, A. Srivastav, and P.
Wehr, Discrepancy of {C}artesian products of arithmetic progressions, Electron.
J. Combin. 11(2004), Research Paper 5, 16 pp.] cannot be extended to more than
colors. In fact, for any and such that does not divide
, there are hypergraphs having arbitrary large discrepancy and
. Apart
from constant factors (depending on and ), in these cases the symmetric
product behaves no better than the general direct product ,
which satisfies .Comment: 12 pages, no figure
Simulation Theorems via Pseudorandom Properties
We generalize the deterministic simulation theorem of Raz and McKenzie
[RM99], to any gadget which satisfies certain hitting property. We prove that
inner-product and gap-Hamming satisfy this property, and as a corollary we
obtain deterministic simulation theorem for these gadgets, where the gadget's
input-size is logarithmic in the input-size of the outer function. This answers
an open question posed by G\"{o}\"{o}s, Pitassi and Watson [GPW15]. Our result
also implies the previous results for the Indexing gadget, with better
parameters than was previously known. A preliminary version of the results
obtained in this work appeared in [CKL+17]
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