Perturbation results on the zero-locus of a polynomial

Let f and g be complex multivariate polynomials of the same degree. Extending Beauzamy's results which hold in the univariate case, we bound the Euclidean distance of points belonging to the zero-loci of f and g in terms of the Bombieri norm of the difference g 12f. We also discuss real perturbations of real polynomials

Homological perturbation theory for nonperturbative integrals

We use the homological perturbation lemma to produce explicit formulas computing the class in the twisted de Rham complex represented by an arbitrary polynomial. This is a non-asymptotic version of the method of Feynman diagrams. In particular, we explain that phenomena usually thought of as particular to asymptotic integrals in fact also occur exactly: integrals of the type appearing in quantum field theory can be reduced in a totally algebraic fashion to integrals over an Euler--Lagrange locus, provided this locus is understood in the scheme-theoretic sense, so that imaginary critical points and multiplicities of degenerate critical points contribute.Comment: 22 pages. Minor revisions from previous versio

Singular perturbation of polynomial potentials in the complex domain with applications to PT-symmetric families

In the first part of the paper, we discuss eigenvalue problems of the form -w"+Pw=Ew with complex potential P and zero boundary conditions at infinity on two rays in the complex plane. We give sufficient conditions for continuity of the spectrum when the leading coefficient of P tends to 0. In the second part, we apply these results to the study of topology and geometry of the real spectral loci of PT-symmetric families with P of degree 3 and 4, and prove several related results on the location of zeros of their eigenfunctions.Comment: The main result on singular perturbation is substantially improved, generalized, and the proof is simplified. 37 pages, 16 figure

Stable Complete Intersections

A complete intersection of n polynomials in n indeterminates has only a finite number of zeros. In this paper we address the following question: how do the zeros change when the coefficients of the polynomials are perturbed? In the first part we show how to construct semi-algebraic sets in the parameter space over which all the complete intersection ideals share the same number of isolated real zeros. In the second part we show how to modify the complete intersection and get a new one which generates the same ideal but whose real zeros are more stable with respect to perturbations of the coefficients.Comment: 1 figur

Multiplicities of Noetherian deformations

The \emph{Noetherian class} is a wide class of functions defined in terms of polynomial partial differential equations. It includes functions appearing naturally in various branches of mathematics (exponential, elliptic, modular, etc.). A conjecture by Khovanskii states that the \emph{local} geometry of sets defined using Noetherian equations admits effective estimates analogous to the effective \emph{global} bounds of algebraic geometry. We make a major step in the development of the theory of Noetherian functions by providing an effective upper bound for the local number of isolated solutions of a Noetherian system of equations depending on a parameter $\epsilon$, which remains valid even when the system degenerates at $\epsilon=0$. An estimate of this sort has played the key role in the development of the theory of Pfaffian functions, and is expected to lead to similar results in the Noetherian setting. We illustrate this by deducing from our main result an effective form of the Lojasiewicz inequality for Noetherian functions.Comment: v2: reworked last section, accepted to GAF

Geometry of free loci and factorization of noncommutative polynomials

The free singularity locus of a noncommutative polynomial f is defined to be the sequence $Z_n(f)=\{X\in M_n^g : \det f(X)=0\}$ of hypersurfaces. The main theorem of this article shows that f is irreducible if and only if $Z_n(f)$ is eventually irreducible. A key step in the proof is an irreducibility result for linear pencils. Apart from its consequences to factorization in a free algebra, the paper also discusses its applications to invariant subspaces in perturbation theory and linear matrix inequalities in real algebraic geometry.Comment: v2: 32 pages, includes a table of content

M Theory Fivebrane and Confining Phase of N=1 SO(Nc)SO(N_c) Gauge Theories

The moduli space of vacua for the confining phase of N=1 $SO(N_c)$ supersymmetric gauge theories in four dimensions is analyzed by studying the M theory fivebrane. The type IIA brane configuration consists of a single NS5 brane, multiple copies of NS'5 branes, D4 branes between them, and D6 branes intersecting D4 branes. We construct M theory fivebrane configuration corresponding to the superpotential perturbation where the power of adjoint field is connected to the number of NS'5 branes. At a singular point in the moduli space where mutually local dyons become massless, the quadratic degeneracy of the N=2 $SO(N_c)$ hyperelliptic curve determines whether this singular point gives a N=1 vacua or not. The comparison of the meson vevs in M theory fivebrane configuration with field theory results turns out that the effective superpotential by the integrating in method with enhanced gauge group SU(2) is exact.Comment: 34 pages, late

Quasi-exactly solvable quartic: elementary integrals and asymptotics

We study elementary eigenfunctions y=p exp(h) of operators L(y)=y"+Py, where p, h and P are polynomials in one variable. For the case when h is an odd cubic polynomial, we found an interesting identity which is used to describe the spectral locus. We also establish some asymptotic properties of the QES spectral locus.Comment: 20 pages, 1 figure. Added Introduction and several references, corrected misprint