Related papers: An iterative approach for real roots of polynomial…

The subject matter of this work is quadratic and cubic polynomial functions with integer coefficients;and all of whose roots are integers. The material of this work is directed primarily at educators,students,and teachers of…

Wooley ({\em J. Number Theory}, 1996) gave an elementary proof of a Bezout like theorem allowing one to count the number of isolated integer roots of a system of polynomial equations modulo some prime power. In this article, we adapt the…

This paper is concerned with certifying that a given point is near an exact root of an overdetermined or singular polynomial system with rational coefficients. The difficulty lies in the fact that consistency of overdetermined systems is…

Many problems in computer algebra and numerical analysis can be reduced to counting or approximating the real roots of a polynomial within an interval. Existing verified root-counting procedures in major proof assistants are mainly based on…

Under certain natural sufficient conditions on the sequence of uniformly bounded closed sets $E_k\subset\mathbb{R}$ of admissible coefficients, we construct a polynomial $P_n(x)=1+\sum_{k=1}^n\varepsilon_k x^k$, $\varepsilon_k\in E_k$, with…

Let $\mathcal{F}_n$ be the set of unitary polynomials of degree $n \ge 2$ that have their roots in $\mathbb{Z}^*$. We note $$ Q(x) := x^n+a_{1}x^{n-1}+\dots+a_{n}. $$ We show that any two fixed consecutive coefficients $(a_{j},a_{j+1})$ ($j…

In this paper, we give some counting results on integer polynomials of fixed degree and bounded height whose distinct non-zero roots are multiplicatively dependent. These include sharp lower bounds, upper bounds and asymptotic formulas for…

We prove that a bivariate polynomial f with exactly t non-zero terms, restricted to a real line {y=ax+b}, either has at most 6t-4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether…

The isolation intervals of the real roots of the real symbolic monic cubic polynomial $p(x) = x^3 + a x^2 + b x + c\,\,$ are found in terms of simple functions of the coefficients of the polynomial (such as: $-a$, $-a/3$, $-c/b$, $\pm…

We give an optimal necessary and sufficient condition for the quotient polynomial and remainder in the division algorithm to have positive coefficients.

This paper investigates the number of monic integer polynomials of degree $n$ whose roots are all real and positive. We establish an asymptotic formula for the case of fixed trace by estimating the number of integer sequences satisfying…

We give formulas for the number of polynomials over a finite field with given root multiplicities, in particular in cases when the formula is surprisingly simple (a power of q). Besides this concrete interpretation, we also prove an…

In some fields such as Mathematics Mechanization, automated reasoning and Trustworthy Computing etc., exact results are needed. Symbolic computations are used to obtain the exact results. Symbolic computations are of high complexity. In…

The existing methods to obtain an exact-fit polynomial does not give the resulting polynomial in its standard form, and further manipulations are needed to obtain that. The new method presented here gives the coefficients of the polynomial…

We consider the family of polynomials $P(x,a)=x^n+a_1x^{n-1}+... +a_n$, $x,a_i\in {\bf R}$, and the stratification of ${\bf R}^n\cong \{(a_1,... ,a_n)|a_i\in {\bf R}\}$ defined by the multiplicity vector of the real roots of $P$. We prove…

We establish new results on root separation of integer, irreducible polynomials of degree at least four. These improve earlier bounds of Bugeaud and Mignotte (for even degree) and of Beresnevich, Bernik, and Goetze (for odd degree).

We obtain a new bound of certain double multiplicative character sums. We use this bound together with some other previously obtained results to obtain new algorithms for finding roots of polynomials modulo a prime $p$.

In this paper, we will derive the real roots of certain sets of matrices with real entries. We will also demonstrate that real orthogonal matrices can have real root or be involutory. Eventually, we will represent idempotent matrices in a…

Suppose $A=\{a_1,\ldots,a_{n+2}\}\subset\mathbb{Z}^n$ has cardinality $n+2$, with all the coordinates of the $a_j$ having absolute value at most $d$, and the $a_j$ do not all lie in the same affine hyperplane. Suppose $F=(f_1,\ldots,f_n)$…

The probability that a zero of a random real polynomial of increasing degree is real tends to zero. However, passing from polynomials to Laurent polynomials yields a surprising result: the probability that a root is real tends not to zero,…