24 research outputs found
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Counting polynomial roots in Isabelle/HOL: A formal proof of the Budan-Fourier theorem
Many problems in computer algebra and numerical analysis can be reduced to counting or approximating the real roots of a polynomial within an interval. Existing verified root-counting procedures in major proof assistants are mainly based on the classical Sturm theorem, which only counts distinct roots.
In this paper, we have strengthened the root-counting ability in Isabelle/HOL by first formally proving the Budan-Fourier theorem. Subsequently, based on Descartes' rule of signs and Taylor shift, we have provided a verified procedure to efficiently over-approximate the number of real roots within an interval, counting multiplicity. For counting multiple roots exactly, we have extended our previous formalisation of Sturm's theorem. Finally, we combine verified components in the developments above to improve our previous certified complex-root-counting procedures based on Cauchy indices. We believe those verified routines will be crucial for certifying programs and building tactics.ERC Advanced Grant ALEXANDRIA (Project 742178
Bifurcation and stability for Nonlinear Schroedinger equations with double well potential in the semiclassical limit
We consider the stationary solutions for a class of Schroedinger equations
with a symmetric double-well potential and a nonlinear perturbation. Here, in
the semiclassical limit we prove that the reduction to a finite-mode
approximation give the stationary solutions, up to an exponentially small term,
and that symmetry-breaking bifurcation occurs at a given value for the strength
of the nonlinear term. The kind of bifurcation picture only depends on the
non-linearity power. We then discuss the stability/instability properties of
each branch of the stationary solutions. Finally, we consider an explicit
one-dimensional toy model where the double well potential is given by means of
a couple of attractive Dirac's delta pointwise interactions.Comment: 46 pages, 4 figure
An integral that counts the zeros of a function
Given a real function on an interval satisfying mild regularity
conditions, we determine the number of zeros of by evaluating a certain
integral. The integrand depends on and . In particular, by
approximating the integral with the trapezoidal rule on a fine enough grid, we
can compute the number of zeros of by evaluating finitely many values of
and . A variant of the integral even allows to determine the number
of the zeros broken down by their multiplicity.Comment: 20 pages, 1 figure, final versio
A census of zeta functions of quartic K3 surfaces over F_2
We compute the complete set of candidates for the zeta function of a K3
surface over F_2 consistent with the Weil conjectures, as well as the complete
set of zeta functions of smooth quartic surfaces over F_2. These sets differ
substantially, but we do identify natural subsets which coincide. This gives
some numerical evidence towards a Honda-Tate theorem for transcendental zeta
functions of K3 surfaces; such a result would refine a recent theorem of
Taelman, in which one must allow an uncontrolled base field extension.Comment: 11 pages; final version, minor changes; to appear in ANTS XI
Descartes's "Rule of Signs'' and Poincar\'e's Positivstellensatz
This is an exposition of Poincar\'e's 1883 paper, ``Sur les \'equations
alg\'ebriques,'' which gives an important refinement of Descartes's rule of
signs and was a precursor of P\' olya's Positivstellensatz.Comment: 6 pages. This report on Poincar\'e's 1883 paper, "Sur les \'equations
alg\'ebriques,'' is based on a talk in the New York Number Theory Seminar on
September 29, 202
Efficiently Computing Real Roots of Sparse Polynomials
We propose an efficient algorithm to compute the real roots of a sparse
polynomial having non-zero real-valued coefficients. It
is assumed that arbitrarily good approximations of the non-zero coefficients
are given by means of a coefficient oracle. For a given positive integer ,
our algorithm returns disjoint disks
, with , centered at the
real axis and of radius less than together with positive integers
such that each disk contains exactly
roots of counted with multiplicity. In addition, it is ensured
that each real root of is contained in one of the disks. If has only
simple real roots, our algorithm can also be used to isolate all real roots.
The bit complexity of our algorithm is polynomial in and , and
near-linear in and , where and constitute
lower and upper bounds on the absolute values of the non-zero coefficients of
, and is the degree of . For root isolation, the bit complexity is
polynomial in and , and near-linear in and
, where denotes the separation of the real roots