The Limited Power of Powering: Polynomial Identity Testing and a Depth-four Lower Bound for the Permanent

Polynomial identity testing and arithmetic circuit lower bounds are two central questions in algebraic complexity theory. It is an intriguing fact that these questions are actually related. One of the authors of the present paper has recently proposed a "real {\tau}-conjecture" which is inspired by this connection. The real {\tau}-conjecture states that the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded. It implies a superpolynomial lower bound on the size of arithmetic circuits computing the permanent polynomial. In this paper we show that the real {\tau}-conjecture holds true for a restricted class of sums of products of sparse polynomials. This result yields lower bounds for a restricted class of depth-4 circuits: we show that polynomial size circuits from this class cannot compute the permanent, and we also give a deterministic polynomial identity testing algorithm for the same class of circuits.Comment: 16 page

On transcendence of numbers related to Sturmian and Arnoux-Rauzy words

We consider numbers of the form Sβ(u) := P∞ n=0 un βn , where u = ⟨un⟩ ∞n=0 is an infinite word over a finite alphabet and β ∈ C satisfies |β| > 1. Our main contribution is to present a combinatorial criterion on u, called echoing, that implies that Sβ(u) is transcendental whenever β is algebraic. We show that every Sturmian word is echoing, as is the Tribonacci word, a leading example of an Arnoux-Rauzy word. We furthermore characterise Q-linear independence of sets of the form {1, Sβ(u1), . . . , Sβ(uk)}, where u1, . . . , uk are Sturmian words having the same slope. Finally, we give an application of the above linear independence criterion to the theory of dynamical systems, showing that for a contracted rotation on the unit circle with algebraic slope, its limit set is either finite or consists exclusively of transcendental elements other than its endpoints 0 and 1. This confirms a conjecture of Bugeaud, Kim, Laurent, and Nogueira

Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya

Preliminaries Consider the local SR-geometry (U, D, g), where U is a neighborhood of 0 ∈ R 3 , D is a Martinet-type distribution, which can be taken in the normal form D = Ker ω, ω = dz − y 2 2 dx, and g is a C ω metric on D, which can be written (see Expanding F 1 and F 2 in Taylor series according to the previous weights and identifying at the order p two elements whose Taylor series are the same at the order p, we obtain the following normal forms of order −1 and 0: • Normal form of order −1: (flat case); • Normal form of order 0: 2 dx 2 + (1 + βx + γy) 2 dy 2 , α, β, γ ∈ R. 1.1. Geodesics equations. The energy minimization problem equivalent to the SR-problem is the following optimal control problem: from Pontryagin's maximum principle [9], minimizing solutions are solutions of the following equations: where H ν is the pseudo-Hamiltonian where ν is a constant normalized to 0 or 1/2. A solution of the previous equations is called an extremal; when ν = 1/2 (resp. ν = 0), the solutions are called normal (resp. abnormal), and their projections onto the state space are called the geodesics. They can be easily computed

Projective geometries arising from Elekes-Szab\'o problems

We generalise the Elekes-Szab\'o theorem to arbitrary arity and dimension and characterise the complex algebraic varieties without power saving. The characterisation involves certain algebraic subgroups of commutative algebraic groups endowed with an extra structure arising from a skew field of endomorphisms. We also extend the Erd\H{o}s-Szemer\'edi sum-product phenomenon to elliptic curves. Our approach is based on Hrushovski's framework of pseudo-finite dimensions and the abelian group configuration theorem.Comment: 48 pages. Minor improvements in presentation. To appear in ASEN