This is the nal report of an internship in algebraic complexity. First, we give an introduction to algebraic complexity and we give some motivations for the study of the main model. Then we present two di erent tools we studied during this internship and use them to establish some lower bounds on this model. We nally discuss whether those bounds could be improved or not.

The Internet of Things (IoT) has become more prevalent in recent years and is an integral part everyday life. Thus, the need for ultralow-power sensors processing units dramatically increased. continuous scaling CMOS transistors resulted severe subthreshold leakage high power density issues. In years, microelectromechanical (MEM) relays have attracted research interests. They are viewed as prom...

Abstract: In this paper, a highly area-efficient multiplier-less FIR filter is presented. Distributed Arithmetic (DA) has been used to implement a bit-serial scheme of a general asymmetric version of an FIR filter, taking optimal advantage of the 3-input LUT-based structure of FPGAs. The implementation of FIR filters on FPGA based on traditional arithmetic method costs considerable hardware res...

This paper presents a VLSI design of a TomlinsonHarashima (TH) precoder for multi-user MIMO (MU-MIMO) systems. The TH precoder consists of LQ decomposition (LQD), interference cancellation (IC), and weight coefficient multiplication (WCM) units. The LQ decomposition unit is based on an application specific instruction-set processor (ASIP) architecture with floating-point arithmetic for high acc...

The class NC of problems solvable by bounded fan-in circuit families of logarithmic depth is known to be contained in logarithmic space L, but not much about the converse is known. In this paper we examine the structure of classes in between NC and L based on counting functions or, equivalently, based on arithmetic circuits. The classes PNC and C=NC, defined by a test for positivity and a test ...

In this paper, we show that there is a family of polynomials {Pn}, where Pn is a polynomial in n variables of degree at most d = O(log2 n), such that • Pn can be computed by linear sized homogeneous depth-5 circuits. • Pn can be computed by poly(n) sized non-homogeneous depth-3 circuits. • Any homogeneous depth-4 circuit computing Pn must have size at least nΩ( √ d). This shows that the paramet...

We study the problem of computing an ensemble of multiple sums where the summands in each sum are indexed by subsets of size p of an n-element ground set. More precisely, the task is to compute, for each subset of size q of the ground set, the sum over the values of all subsets of size p that are disjoint from the subset of size q. We present an arithmetic circuit that, without subtraction, sol...

According to the real τ -conjecture, the number of real roots of a sum of products of sparse univariate polynomials should be polynomially bounded in the size of such an expression. It is known that this conjecture implies a superpolynomial lower bound on the arithmetic circuit complexity of the permanent. In this paper, we use the Wronksian determinant to give an upper bound on the number of r...

In almost all of the currently working circuits, especially in analog circuits implementing signal processing applications, basic arithmetic operations such as multiplication, addition, subtraction and division are performed on values which are represented by voltages or currents. However, in this paper, we propose a new and simple method for performing analog arithmetic operations which in thi...