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

Meandering of trajectories of polynomial vector fields in the affine n-space

We give an explicit upper bound for the number of isolated intersections between an integral curve of a polynomial vector field in Rn and an affine hyperplane. The problem turns out to be closely related to finding an explicit upper bound for the length of ascending chains of polynomial ideals spanned by consecutive derivatives. This exposition constitutes an extended abstract of a forthcoming paper: only the basic steps are outlined here, with all technical details being either completely omitted or at best indicated

Multiplicity Estimates: a Morse-theoretic approach

The problem of estimating the multiplicity of the zero of a polynomial when restricted to the trajectory of a non-singular polynomial vector field, at one or several points, has been considered by authors in several different fields. The two best (incomparable) estimates are due to Gabrielov and Nesterenko. In this paper we present a refinement of Gabrielov's method which simultaneously improves these two estimates. Moreover, we give a geometric description of the multiplicity function in terms certain naturally associated polar varieties, giving a topological explanation for an asymptotic phenomenon that was previously obtained by elimination theoretic methods in the works of Brownawell, Masser and Nesterenko. We also give estimates in terms of Newton polytopes, strongly generalizing the classical estimates.Comment: Minor revision; To appear in Duke Math. Journa

Multiplicity estimates, analytic cycles and Newton polytopes

We consider the problem of estimating the multiplicity of a polynomial when restricted to the smooth analytic trajectory of a (possibly singular) polynomial vector field at a given point or points, under an assumption known as the D-property. Nesterenko has developed an elimination theoretic approach to this problem which has been widely used in transcendental number theory. We propose an alternative approach to this problem based on more local analytic considerations. In particular we obtain simpler proofs to many of the best known estimates, and give more general formulations in terms of Newton polytopes, analogous to the Bernstein-Kushnirenko theorem. We also improve the estimate's dependence on the ambient dimension from doubly-exponential to an essentially optimal single-exponential.Comment: Some editorial modifications to improve readability; No essential mathematical change

Perception Based Navigation for Underactuated Robots.

Robot autonomous navigation is a very active field of robotics. In this thesis we propose a hierarchical approach to a class of underactuated robots by composing a collection of local controllers with well understood domains of attraction. We start by addressing the problem of robot navigation with nonholonomic motion constraints and perceptual cues arising from onboard visual servoing in partially engineered environments. We propose a general hybrid procedure that adapts to the constrained motion setting the standard feedback controller arising from a navigation function in the fully actuated case. This is accomplished by switching back and forth between moving "down" and "across" the associated gradient field toward the stable manifold it induces in the constrained dynamics. Guaranteed to avoid obstacles in all cases, we provide conditions under which the new procedure brings initial configurations to within an arbitrarily small neighborhood of the goal. We summarize with simulation results on a sample of visual servoing problems with a few different perceptual models. We document the empirical effectiveness of the proposed algorithm by reporting the results of its application to outdoor autonomous visual registration experiments with the robot RHex guided by engineered beacons. Next we explore the possibility of adapting the resulting first order hybrid feedback controller to its dynamical counterpart by introducing tunable damping terms in the control law. Just as gradient controllers for standard quasi-static mechanical systems give rise to generalized "PD-style" controllers for dynamical versions of those standard systems, we show that it is possible to construct similar "lifts" in the presence of non-holonomic constraints notwithstanding the necessary absence of point attractors. Simulation results corroborate the proposed lift. Finally we present an implementation of a fully autonomous navigation application for a legged robot. The robot adapts its leg trajectory parameters by recourse to a discrete gradient descent algorithm, while managing its experiments and outcome measurements autonomously via the navigation visual servoing algorithms proposed in this thesis.Ph.D.Electrical Engineering: SystemsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/58412/1/glopes_1.pd