The fermionant Fermn(x̄) = ∑ σ∈Sn (−k)c(π) ∏n i=1 xi,j can be seen as a generalization of both the permanent (for k = −1) and the determinant (for k = 1). We demonstrate that it is VNP-complete for any rational k , 1. Furthermore it is #P -complete for the same values of k. The immanant is also a generalization of the permanent (for a Young diagram with a single line) and of the determinant (whe...

We introduce the polynomial coefficient matrix and identify maximum rank of this matrix under variable substitution as a complexity measure for multivariate polynomials. We use our techniques to prove super-polynomial lower bounds against several classes of non-multilinear arithmetic circuits. In particular, we obtain the following results : • As our main result, we prove that any homogeneous d...

This paper provide the principles of Modified Distributed Arithmetic, and introduce it into the FIR filters design, and then presents a 31-order FIR low-pass filter using Modified Distributed Arithmetic, which save considerable MAC blocks to decrease the circuit scale, meanwhile, divided LUT method is used to decrease the required memory units and pipeline structure is also used to increase the...

‎we obtain the asymptotic expansion of the sequence with general term $frac{a_n}{g_n}$‎, ‎where $a_n$ and $g_n$ are the arithmetic and geometric means of the numbers $d(1),d(2),dots,d(n)$‎, ‎with $d(n)$ denoting the number of positive divisors of $n$‎. ‎also‎, ‎we obtain some explicit bounds concerning $g_n$ and $frac{a_n}{g_n}$.

Let τ(n) denote the minimum number of arithmetic operations sufficient to build the integer n from the constant 1. We prove that if there are arithmetic circuits for computing the permanent of n by n matrices having size polynomial in n, then τ(n!) is polynomially bounded in log n. Under the same assumption on the permanent, we conclude that the Pochhammer-Wilkinson polynomials ∏n k=1(X − k) an...

The motivation for this paper is to study the complexity of constant-width arithmetic circuits. Our main results are the following. 1. For every k > 1, we provide an explicit polynomial that can be computed by a linear-sized monotone circuit of width 2k but has no subexponential-sized monotone circuit of width k. It follows, from the definition of the polynomial, that the constant-width and the...

In their paper on the “chasm at depth four”, Agrawal and Vinay have shown that polynomials in m variables of degree O(m) which admit arithmetic circuits of size 2 also admit arithmetic circuits of depth four and size 2. This theorem shows that for problems such as arithmetic circuit lower bounds or black-box derandomization of identity testing, the case of depth four circuits is in a certain sense

An arithmetic circuit is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest subalgebra of the complex algebra of the semiring of natural numbers. In the present paper we investigate the algebraic structure of complex algebras of natural num...