172 research outputs found
On a tropical dual Nullstellensatz
Since a tropical Nullstellensatz fails even for tropical univariate
polynomials we study a conjecture on a tropical {\it dual} Nullstellensatz for
tropical polynomial systems in terms of solvability of a tropical linear system
with the Cayley matrix associated to the tropical polynomial system. The
conjecture on a tropical effective dual Nullstellensatz is proved for tropical
univariate polynomials
Tropical differential equations
Tropical differential equations are introduced and an algorithm is designed
which tests solvability of a system of tropical linear differential equations
within the complexity polynomial in the size of the system and in its
coefficients. Moreover, we show that there exists a minimal solution, and the
algorithm constructs it (in case of solvability). This extends a similar
complexity bound established for tropical linear systems. In case of tropical
linear differential systems in one variable a polynomial complexity algorithm
for testing its solvability is designed.
We prove also that the problem of solvability of a system of tropical
non-linear differential equations in one variable is -hard, and this
problem for arbitrary number of variables belongs to . Similar to tropical
algebraic equations, a tropical differential equation expresses the (necessary)
condition on the dominant term in the issue of solvability of a differential
equation in power series
Probabilistic communication complexity over the reals
Deterministic and probabilistic communication protocols are introduced in
which parties can exchange the values of polynomials (rather than bits in the
usual setting). It is established a sharp lower bound on the communication
complexity of recognizing the -dimensional orthant, on the other hand the
probabilistic communication complexity of its recognizing does not exceed 4. A
polyhedron and a union of hyperplanes are constructed in \RR^{2n} for which a
lower bound on the probabilistic communication complexity of recognizing
each is proved. As a consequence this bound holds also for the EMPTINESS and
the KNAPSACK problems
Authentication from matrix conjugation
We propose an authentication scheme where forgery (a.k.a. impersonation)
seems infeasible without finding the prover's long-term private key. The latter
would follow from solving the conjugacy search problem in the platform
(noncommutative) semigroup, i.e., to recovering X from X^{-1}AX and A. The
platform semigroup that we suggest here is the semigroup of nxn matrices over
truncated multivariable polynomials over a ring.Comment: 6 page
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