Boundaries of VP and VNP

One fundamental question in the context of the geometric complexity theory approach to the VP vs. VNP conjecture is whether VP = VP, where VP is the class of families of polynomials that can be computed by arithmetic circuits of polynomial degree and size, and VP is the class of families of polynomials that can be approximated infinitesimally closely by arithmetic circuits of polynomial degree and size. The goal of this article is to study the conjecture in (Mulmuley, FOCS 2012) that VP is not contained in VP. Towards that end, we introduce three degenerations of VP (i.e., sets of points in VP), namely the stable degeneration Stable-VP, the Newton degeneration Newton-VP, and the p-definable one-parameter degeneration VP∗. We also introduce analogous degenerations of VNP. We show that Stable-VP ⊆ Newton-VP ⊆ VP∗ ⊆ VNP, and Stable-VNP = Newton-VNP = VNP∗ = VNP. The three notions of degenerations and the proof of this result shed light on the problem of separating VP from VP. Although we do not yet construct explicit candidates for the polynomial families in VP \VP, we prove results which tell us where not to look for such families. Specifically, we demonstrate that the families in Newton-VP \VP based on semi-invariants of quivers would have to be nongeneric by showing that, for many finite quivers (including some wild ones), Newton degeneration of any generic semi-invariant can be computed by a circuit of polynomial size. We also show that the Newton degenerations of perfect matching Pfaffians, monotone arithmetic circuits over the reals, and Schur polynomials have polynomial-size circuits

P versus NP and geometry

I describe three geometric approaches to resolving variants of P v. NP, present several results that illustrate the role of group actions in complexity theory, and make a first step towards completely geometric definitions of complexity classes.Comment: 20 pages, to appear in special issue of J. Symbolic. Comp. dedicated to MEGA 200

Discovering the roots: Uniform closure results for algebraic classes under factoring

Newton iteration (NI) is an almost 350 years old recursive formula that approximates a simple root of a polynomial quite rapidly. We generalize it to a matrix recurrence (allRootsNI) that approximates all the roots simultaneously. In this form, the process yields a better circuit complexity in the case when the number of roots $r$ is small but the multiplicities are exponentially large. Our method sets up a linear system in $r$ unknowns and iteratively builds the roots as formal power series. For an algebraic circuit $f(x_1,\ldots,x_n)$ of size $s$ we prove that each factor has size at most a polynomial in: $s$ and the degree of the squarefree part of $f$. Consequently, if $f_1$ is a $2^{\Omega(n)}$-hard polynomial then any nonzero multiple $\prod_{i} f_i^{e_i}$ is equally hard for arbitrary positive $e_i$'s, assuming that $\sum_i \text{deg}(f_i)$ is at most $2^{O(n)}$. It is an old open question whether the class of poly($n$)-sized formulas (resp. algebraic branching programs) is closed under factoring. We show that given a polynomial $f$ of degree $n^{O(1)}$ and formula (resp. ABP) size $n^{O(\log n)}$ we can find a similar size formula (resp. ABP) factor in randomized poly($n^{\log n}$)-time. Consequently, if determinant requires $n^{\Omega(\log n)}$ size formula, then the same can be said about any of its nonzero multiples. As part of our proofs, we identify a new property of multivariate polynomial factorization. We show that under a random linear transformation $\tau$, $f(\tau\overline{x})$ completely factors via power series roots. Moreover, the factorization adapts well to circuit complexity analysis. This with allRootsNI are the techniques that help us make progress towards the old open problems, supplementing the large body of classical results and concepts in algebraic circuit factorization (eg. Zassenhaus, J.NT 1969, Kaltofen, STOC 1985-7 \& Burgisser, FOCS 2001).Comment: 33 Pages, No figure

Monotone Projection Lower Bounds from Extended Formulation Lower Bounds

In this short note, we reduce lower bounds on monotone projections of polynomials to lower bounds on extended formulations of polytopes. Applying our reduction to the seminal extended formulation lower bounds of Fiorini, Massar, Pokutta, Tiwari, & de Wolf (STOC 2012; J. ACM, 2015) and Rothvoss (STOC 2014; J. ACM, 2017), we obtain the following interesting consequences. 1. The Hamiltonian Cycle polynomial is not a monotone subexponential-size projection of the permanent; this both rules out a natural attempt at a monotone lower bound on the Boolean permanent, and shows that the permanent is not complete for non-negative polynomials in VNP$_{{\mathbb R}}$ under monotone p-projections. 2. The cut polynomials and the perfect matching polynomial (or "unsigned Pfaffian") are not monotone p-projections of the permanent. The latter, over the Boolean and-or semi-ring, rules out monotone reductions in one of the natural approaches to reducing perfect matchings in general graphs to perfect matchings in bipartite graphs. As the permanent is universal for monotone formulas, these results also imply exponential lower bounds on the monotone formula size and monotone circuit size of these polynomials.Comment: Published in Theory of Computing, Volume 13 (2017), Article 18; Received: November 10, 2015, Revised: July 27, 2016, Published: December 22, 201

Solar cycle variations of the Cluster spacecraft potential and its use for electron density estimations

International audience[1] A sunlit conductive spacecraft, immersed in tenuous plasma, will attain a positive potential relative to the ambient plasma. This potential is primarily governed by solar irradiation, which causes escape of photoelectrons from the surface of the spacecraft, and the electrons in the ambient plasma providing the return current. In this paper we combine potential measurements from the Cluster satellites with measurements of extreme ultraviolet radiation from the TIMED satellite to establish a relation between solar radiation and spacecraft charging from solar maximum to solar minimum. We then use this relation to derive an improved method for determination of the current balance of the spacecraft. By calibration with other instruments we thereafter derive the plasma density. The results show that this method can provide information about plasma densities in the polar cap and magnetotail lobe regions where other measurements have limitations

Critical angular velocity for vortex lines formation

For helium II inside a rotating cylinder, it is proposed that the formation of vortex lines of the frictionless superfluid component of the liquid is caused by the presence of the rotating quasi-particles gas. By minimising the free energy of the system, the critical value Omega_0 of the angular velocity for the formation of the first vortex line is determined. This value nontrivially depends on the temperature, and numerical estimations of its temperature behaviour are produced. It is shown that the latent heat for a vortex formation and the associated discontinuous change in the angular momentum of the quasi-particles gas determine the slope of Omega_0 (T) via some kind of Clapeyron equation.Comment: 16 page