Related papers: Progress on Polynomial Identity Testing - II
NP complete problem is one of the most challenging issues. The question of whether all problems in NP are also in P is generally considered one of the most important open questions in mathematics and theoretical computer science as it has…
This paper presents an alternative approach to simplify the proofs of some important results related to polynomial mappings in Computational Algebraic Geometry such as Polynomial Implicitization, Image Closure and some properties of the…
Complexity is a multi-faceted phenomenon, involving a variety of features including disorder, nonlinearity, and self-organisation. We use a recently developed rigorous framework for complexity to understand measures of complexity. We…
Powerful skew arithmetic circuits are introduced. These are skew arithmetic circuits with variables, where input gates can be labelled with powers $x^n$ for binary encoded numbers $n$. It is shown that polynomial identity testing for…
In this paper, we study properties of polynomials over division rings. Moreover, we present formulas for finding roots of some polynomials
Given two testable properties $\mathcal{P}_{1}$ and $\mathcal{P}_{2}$, under what conditions are the union, intersection or set-difference of these two properties also testable? We initiate a systematic study of these basic set-theoretic…
The different notions of matings of pairs of equal degree polynomials are introduced and are related to each other as well as known results on matings. The possible obstructions to matings are identified and related. Moreover the relations…
Motivated by the fundamental lower bounds questions in proof complexity, we initiate the study of matrix identities as hard instances for strong proof systems. A matrix identity of $d \times d$ matrices over a field $\mathbb{F}$, is a…
We survey results on the formalization and independence of mathematical statements related to major open problems in computational complexity theory. Our primary focus is on recent findings concerning the (un)provability of complexity…
We study the problem of obtaining deterministic black-box polynomial identity testing algorithms (PIT) for algebraic branching programs (ABPs) that are read-once and oblivious. This class has an deterministic white-box polynomial identity…
Additive Fourier Transform is sdudied. A fast multiplication algorithm for polynomials over the binary field is given. The bit complexity of the algorithm is $O(n(log n)(\log\log n)^2)$.
We investigate a series of learning kernel problems with polynomial combinations of base kernels, which will help us solve regression and classification problems. We also perform some numerical experiments of polynomial kernels with…
The main result of this paper is to show that all binomial identities are orderable. This is a natural statement in the combinatorial theory of finite sets, which can also be applied in distributed computing to derive new strong bounds on…
The area of property testing tries to design algorithms that can efficiently handle very large amounts of data: given a large object that either has a certain property or is somehow "far" from having that property, a tester should…
A family of general integral identities is derived and several applications of physical interest are presented
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
We analyse the complexity of solving the discrete logarithm problem and of testing the principality of ideals in a certain class of number fields. We achieve the subexponential complexity in $O(L(1/3,O(1)))$ when both the discriminant and…
In this paper, we obtain a generalization of an identity due to Carlitz on Bernoulli polynomials. Then we use this generalized formula to derive two symmetric identities which reduce to some known identities on Bernoulli polynomials and…
One approach to make progress on the symbolic determinant identity testing (SDIT) problem is to study the structure of singular matrix spaces. After settling the non-commutative rank problem (Garg-Gurvits-Oliveira-Wigderson, Found. Comput.…
Despite encouraging recent progresses in ensemble approaches, classification methods seem to have reached a plateau in development. Further advances depend on a better understanding of geometrical and topological characteristics of point…