Related papers: A note on Cardano's formula
Analysing the cubic sectors of a real polynomial of degree n, a modification of the Newton Rule is Signs is proposed with which stricter upper bound on the number of real roots can be found. A new necessary condition for reality of the…
We suggest a closed solution for the roots of polynomial trinomial algebraic equation $$z^n+xz^{n-1}-1=0$$ with an appropriate $x$. This solution is a minor modification to the work of Mikhalkin (Mikhalkin E N, 2006. On solving general…
A method of constructing specific polynomial representations $f(x)$ over the finite field $\mathbb{F}_p$ of the square roots function modulo a prime $p = 2^kn + 1$, $n$ odd, is presented. The formulas for the cases $k = 2$, $3$ and $4$ are…
We provide sufficient conditions for systems of polynomial equations over general (real or complex) algebras to have a solution. This generalizes known results on quaternions, octonions and matrix algebras. We also generalize the…
In a previous article we have proved non-existence of certain "solutions" of the cubically nonlinear Schr\"odinger equation in the general case, and presented solutions in the non-generic case. -- In the present article we describe a…
For each positive integer n greater than or equal to 2, a new approach to expressing real numbers as sequences of nonnegative integers is given. The n=2 case is equivalent to the standard continued fraction algorithm. For n=3, it reduces to…
Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.
This note studies, and partially solves, 3 elementary questions about continuous rational functions on real (and p-adic) algebraic varieties: Can one restrict such a function to a subvariety? Can one extend such a function from a…
We present a new algorithm for refining a real interval containing a single real root: the new method combines characteristics of the classical Bisection algorithm and Newton's Iteration. Our method exhibits quadratic convergence when…
Our main result is that any real cubic algebraic number has a continued fraction expansion with polynomial coefficients. Some generalizations are mentioned.
Presented is a two-tier analysis of the location of the real roots of the general quartic equation $x^4 + ax^3 + bx^2 + cx + d = 0$ with real coefficients and the classification of the roots in terms of $a$, $b$, $c$, and $d$, without using…
Exact solutions with the initial conditions are presented in the cubic duffing equation. These exact solutions are expressed in terms of the leaf function and the trigonometric function. The leaf functions: $r=sleaf_n(t) $ or $…
In this paper we report on an application of computer algebra in which mathematical puzzles are generated of a type that had been widely used in mathematics contests by a large number of participants worldwide. The algorithmic aspect of our…
In this work, we describe our experience in learning the use of a computer proof assistant - specifically, Lean - from scratch, through proving formulae for the solutions of polynomial equations. Specifically, in this work we characterize…
This article provides a simple geometric interpretation of the quadratic formula. The geometry helps to demystify the formula's complex appearance and casts it into a much simpler existence, thus potentially benefits early algebra students.
We investigate the average number of solutions of certain quadratic congruences. As an application, we establish Manin's conjecture for a cubic surface whose singularity type is A_5+A_1.
In this paper, the compact linearization approach originally proposed for binary quadratic programs with assignment constraints is generalized to such programs with arbitrary linear equations and inequalities that have positive coefficients…
Heron, in Metrica III.20-22, is concerned with the the division of solid figures - pyramids, cones and frustra of cones - to which end there is a need to extract cube roots. We report here on some of our findings on the conjecture by…
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals. The bounds are uniform in the curve and involve the rank of the corresponding Jacobian. The method used in the proof is a…
We prove that all arrangements (consistent with the Rolle theorem and some other natural restrictions) of the real roots of a real polynomial and of its $s$-th derivative are realizable by real polynomials.