570 research outputs found
On the Topologies on ind-Varieties and related Irreducibility Questions
In the literature there are two ways of endowing an affine ind-variety with a
topology. One possibility is due to Shafarevich and the other to Kambayashi. In
this paper we specify a large class of affine ind-varieties where these two
topologies differ. We give an example of an affine ind-variety that is
reducible with respect to Shafarevich's topology, but irreducible with respect
to Kambayashi's topology. Moreover, we give a counter-example of a supposed
irreducibility criterion of Shafarevich which is different from a
counter-example given by Homma. We finish the paper with an irreducibility
criterion similar to the one given by Shafarevich.Comment: 11 pages, typos corrected, minor changes, improved expositio
Construction of Rational Surfaces Yielding Good Codes
In the present article, we consider Algebraic Geometry codes on some rational
surfaces. The estimate of the minimum distance is translated into a point
counting problem on plane curves. This problem is solved by applying the upper
bound "\`a la Weil" of Aubry and Perret together with the bound of Homma and
Kim for plane curves. The parameters of several codes from rational surfaces
are computed. Among them, the codes defined by the evaluation of forms of
degree 3 on an elliptic quadric are studied. As far as we know, such codes have
never been treated before. Two other rational surfaces are studied and very
good codes are found on them. In particular, a [57,12,34] code over
and a [91,18,53] code over are discovered, these
codes beat the best known codes up to now.Comment: 20 pages, 7 figure
Modified Makar-Limanov and Derksen invariants
We investigate modified Makar-Limanov and Derksen invariants of an affine
algebraic variety. The modified Makar-Limanov invariant is the intersection of
kernels of all locally nilpotent derivations with slices and the modified
Derksen invariant is the subalgebra generated by these kernels. We prove that
modified Makar-Limanov invariant coincide with Makar-Limanov invariant if there
exists a locally nilpotent derivation with a slice. Also we construct an
example of a variety admitting a locally nilpotent derivation with a slice such
that modified Derksen invariant does not coincide with Derksen invariant
The group of automorphisms of the first weyl algebra in prime characteristic and the restriction map
Let K be a perfect field of characteristic p > 0; A(1) := K be the first Weyl algebra; and Z := K[X := x(p), Y := partial derivative(p)] be its centre. It is proved that (1) the restriction map res : Aut(K)(A(1)) -> Aut(K)(Z), sigma bar right arrow sigma vertical bar(Z) is a monomorphism with im(res) = Gamma := (tau is an element of Aut(K)(Z) vertical bar J(tau) = 1), where J(tau) is the Jacobian of tau, (Note that Aut(K)(Z) = K* (sic) Gamma, and if K is not perfect then im(res) not equal Gamma.); (ii) the bijection res : Aut(K)(A(1)) -> Gamma is a monomorphism of infinite dimensional algebraic groups which is not an isomorphism (even if K is algebraically closed); (iii) an explicit formula for res(-1) is found via differential operators D(Z) on Z and negative powers of the Fronenius map F. Proofs are based on the following (non-obvious) equality proved in the paper: (d/dx + f)(p) = (d/dx)(p) + d(p-1)f/dx(p-1) + f(p), f is an element of K[x]
A PSPACE Construction of a Hitting Set for the Closure of Small Algebraic Circuits
In this paper we study the complexity of constructing a hitting set for the
closure of VP, the class of polynomials that can be infinitesimally
approximated by polynomials that are computed by polynomial sized algebraic
circuits, over the real or complex numbers. Specifically, we show that there is
a PSPACE algorithm that given n,s,r in unary outputs a set of n-tuples over the
rationals of size poly(n,s,r), with poly(n,s,r) bit complexity, that hits all
n-variate polynomials of degree-r that are the limit of size-s algebraic
circuits. Previously it was known that a random set of this size is a hitting
set, but a construction that is certified to work was only known in EXPSPACE
(or EXPH assuming the generalized Riemann hypothesis). As a corollary we get
that a host of other algebraic problems such as Noether Normalization Lemma,
can also be solved in PSPACE deterministically, where earlier only randomized
algorithms and EXPSPACE algorithms (or EXPH assuming the generalized Riemann
hypothesis) were known.
The proof relies on the new notion of a robust hitting set which is a set of
inputs such that any nonzero polynomial that can be computed by a polynomial
size algebraic circuit, evaluates to a not too small value on at least one
element of the set. Proving the existence of such a robust hitting set is the
main technical difficulty in the proof.
Our proof uses anti-concentration results for polynomials, basic tools from
algebraic geometry and the existential theory of the reals
On the automorphism group of a toral variety
Let be an algebraically closed field of characteristic zero. An
affine algebraic variety over is toral if it is isomorphic to
a closed subvariety of a torus . We study the group
of regular automorpshims of a toral variety . We prove
that if is a maximal torus in , then is a direct
product , where is a toral variety with a trivial maximal torus
in the automorphism group. We show that knowing , one can
compute . In the case when the rank of the group
is , the group can
be described explicitly
Computing Small Certificates of Inconsistency of Quadratic Fewnomial Systems
B{\'e}zout 's theorem states that dense generic systems of n multivariate
quadratic equations in n variables have 2 n solutions over algebraically closed
fields. When only a small subset M of monomials appear in the equations
(fewnomial systems), the number of solutions may decrease dramatically. We
focus in this work on subsets of quadratic monomials M such that generic
systems with support M do not admit any solution at all. For these systems,
Hilbert's Nullstellensatz ensures the existence of algebraic certificates of
inconsistency. However, up to our knowledge all known bounds on the sizes of
such certificates -including those which take into account the Newton polytopes
of the polynomials- are exponential in n. Our main results show that if the
inequality 2|M| -- 2n \sqrt 1 + 8{\nu} -- 1 holds for a quadratic
fewnomial system -- where {\nu} is the matching number of a graph associated
with M, and |M| is the cardinality of M -- then there exists generically a
certificate of inconsistency of linear size (measured as the number of
coefficients in the ground field K). Moreover this certificate can be computed
within a polynomial number of arithmetic operations. Next, we evaluate how
often this inequality holds, and we give evidence that the probability that the
inequality is satisfied depends strongly on the number of squares. More
precisely, we show that if M is picked uniformly at random among the subsets of
n + k + 1 quadratic monomials containing at least (n 1/2+)
squares, then the probability that the inequality holds tends to 1 as n grows.
Interestingly, this phenomenon is related with the matching number of random
graphs in the Erd{\"o}s-Renyi model. Finally, we provide experimental results
showing that certificates in inconsistency can be computed for systems with
more than 10000 variables and equations.Comment: ISSAC 2016, Jul 2016, Waterloo, Canada. Proceedings of ISSAC 201
SEMICLASSICAL ASYMPTOTICS OF EIGENVALUES FOR NON-SELFADJOINT OPERATORS AND QUANTIZATION CONDITIONS ON RIEMANN SURFACES
This paper reports a study of the semiclassical asymptotic behavior of the eigenvalues of some nonself-adjoint operators that are important for applications. These operators are the Schrödinger operator with complex periodic potential and the operator of induction. It turns out that the asymptotics of the spectrum can be calculated using the quantization conditions. These can be represented as the condition that the integrals of a holomorphic form over the cycles on the corresponding complex Lagrangian manifold, which is a Riemann surface of constant energy, are integers. In contrast to the real case (the BohrâSommerfeldâMaslov formulas), in order to calculate a chosen spectral series, it is sufficient to assume that the integral over only one of the cycles takes integer values, and different cycles determine different parts of the spectrum
- âŠ