Related papers: Solving Polynomial Equations from Complex Numbers
We describe a classification of degree n complex coefficient polynomials with respect to combinatorial patterns that arise from the two real algebraic curves obtained as the zero sets for their real and imaginary part. In particular, we…
We consider the notions of (i) critical points, (ii) second-order points, (iii) local minima, and (iv) strict local minima for multivariate polynomials. For each type of point, and as a function of the degree of the polynomial, we study the…
We develop new polynomial methods for studying systems of word equations. We use them to improve some earlier results and to analyze how sizes of systems of word equations satisfying certain independence properties depend on the lengths of…
The determinantal complexity of a polynomial $P \in \mathbb{F}[x_1, \ldots, x_n]$ over a field $\mathbb{F}$ is the dimension of the smallest matrix $M$ whose entries are affine functions in $\mathbb{F}[x_1, \ldots, x_n]$ such that $P =…
We provide an implicit characterization of polynomial time computation in terms of ordinary differential equations: we characterize the class $\operatorname{PTIME}$ of languages computable in polynomial time in terms of differential…
We consider the problem of bounding away from 0 the minimum value m taken by a polynomial P of Z[X_1,...,X_k] over the standard simplex, assuming that m>0. Recent algorithmic developments in real algebraic geometry enable us to obtain a…
We prove a complex polynomial of degree $n$ has at most $\lceil n/2 \rceil$ attractive fixed points lying on a line. We also consider the general case.
The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height…
In this paper we give a new and simple algorithm to put any multivariate polynomial into a normal determinant form in which each entry has the form , and in each column the same variable appears. We also apply the algorithm to obtain a…
We give some real polynomials in two variables of degrees 4, 5, and 6 whose hessian curves have more connected components than had been known previously. In particular, we give a quartic polynomial whose hessian curve has 4 compact…
We look at the number of solutions of an equation of the form f_1*f_2*...*f_k=a in a finite field, where each f_i is a multilinear polynomial. We use two methods to construct a solution of this problem for the cases a=0, a<>0, and we…
A system of commutative complex numbers in 5 dimensions of the form u=x_0+h_1x_1+h_2x_2+h_3x_3+h_4x_4 is described in this paper, the variables x_0, x_1, x_2, x_3, x_4 being real numbers. The operations of addition and multiplication of the…
In Physics, we are generally interested in real solutions involving natural phenomena, where knowledge of real functions of real variables is sufficient to obtain physically relevant results. However, the complexity of phenomena associated…
We consider systems of strict multivariate polynomial inequalities over the reals. All polynomial coefficients are parameters ranging over the reals, where for each coefficient we prescribe its sign. We are interested in the existence of…
We present a new algorithm for reconstructing an exact algebraic number from its approximate value using an improved parameterized integer relation construction method. Our result is consistent with the existence of error controlling on…
Using the same method we provide negative answers to the following questions: Is it possible to find real equations for complex polynomials in two variables up to topological equivalence (Lee Rudolph) ? Can two topologically equivalent…
We present a complete algorithm for finding an exact minimal polynomial from its approximate value by using an improved parameterized integer relation construction method. Our result is superior to the existence of error controlling on…
The method of this paper is my original creation. A new method for solving linear differential equations is proposed in this paper. The important conclusion of this paper is that arbitrary order linear ordinary differential equations with…
Given any polynomial with real coefficients, the existence of a real quadratic polynomial factor is proven using only basic real analysis. The aim is to provide an approachable proof to anybody who is familiar with the least upper bound…
The Collatz conjecture is explored using polynomials based on a binary numeral system. It is shown that the degree of the polynomials, on average, decreases after a finite number of steps of the Collatz operation, which provides a weak…