We present a complete numerical algorithm of isolating all the real zeros of a zero-dimensional triangular polynomial system Fn ⊆ Z[x1, . . . , xn]. Our system Fn is general, with no further assumptions. In particular, our algorithm successfully treat multiple zeros directly in such systems. A key idea is to introduce evaluation bounds and sleeve bounds. We also present a much more efficient al...

Bernstein polynomials have been recently used for the solution of some linear and non-linear differential equations, both partial and ordinary, by Bhatta and Bhatti [1] and Bhatti and Bracken [2]. Also these have been used to solve some classes of inegral equations of both first and second kinds, by Mandal and Bhattacharya [3]. These were further used to solve a Cauchy singular integro-differen...

We propose public-key cryptosystems with public key a system of polynomial equations and private key an ideal.

For a generic n-degree polynomial system which contains n+1 polynomials in n variables, we give the construction of the generic mixed Cayley-Sylvester resultant matrix. There are n− 1 generic mixed Cayley-Sylvester resultant matrices between the classical Cayley resultant matrix and the classical Sylvester resultant matrix. The entries of these new resultant matrix are of degree m(1 < m < n + 1...

As already done for the matrix case in [6, p.256], [1, Thm. 6.1, p.1872] and [10, Thm. 3.2] we give a parametrization of the Bouligand tangent cone of the variety of tensors of bounded TT rank. We discuss how the proof generalizes to any binary hierarchical format. The parametrization can be rewritten as an orthogonal sum of TT tensors. Its retraction onto the variety is particularly easy to co...

We present the Zhuang-Zi algorithm, a new method for solving multivariate polynomial equations over a finite field. We describe the algorithm and present examples, some of which cannot be solved with the fastest known algorithms.

The problem considered in this paper is the computation of all solutions of a given polynomial system in a bounded domain. Proving Rouché’s Theorem by homotopy continuation concepts yields a new class of homotopy methods, the socalled regional homotopy methods. These methods rely on isolating a part of the system to be solved, which dominates the rest of the system on the border of the domain. ...

How many operations do we need on the average to compute an approximate root of a random Gaussian polynomial system? Beyond Smale’s 17th problem that asked whether a polynomial bound is possible, we prove a quasi-optimal bound (input size)1`op1q. This improves upon the previously known (input size) 3 2 `op1q bound. The new algorithm relies on numerical continuation along rigid continuation path...

Balas introduced intersection cuts for mixed integer linear sets. Intersection cuts are given by closed form formulas and form an important class of cuts for solving mixed integer linear programs. In this paper we introduce an extension of intersection cuts to mixed integer conic quadratic sets. We identify the formula for the conic quadratic intersection cut by formulating a system of polynomi...

We propose a new space efficient operator to multiply elements lying in a binary field F2k . Our approach is based on a novel system of representation called Double Polynomial System which set elements as a bivariate polynomials over F2. Thanks to this system of representation, we are able to use a Lagrange representation of the polynomials and then get a logarithmic time multiplier with a spac...