Related papers: Root Separation for Trinomials

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…

Suppose f is a real univariate polynomial of degree D with exactly 4 monomial terms. We present an algorithm, with complexity polynomial in log D on average (relative to the stable log-uniform measure), for counting the number of real roots…

We consider the problem of approximating all real roots of a square-free polynomial $f$. Given isolating intervals, our algorithm refines each of them to a width of $2^{-L}$ or less, that is, each of the roots is approximated to $L$ bits…

Suppose $q$ is a prime power and $f\in\mathbb{F}_q[x]$ is a univariate polynomial with exactly $t$ monomial terms and degree $<q-1$. To establish a finite field analogue of Descartes' Rule, Bi, Cheng, and Rojas (2013) proved an upper bound…

We propose an efficient algorithm to compute the real roots of a sparse polynomial $f\in\mathbb{R}[x]$ having $k$ non-zero real-valued coefficients. It is assumed that arbitrarily good approximations of the non-zero coefficients are given…

We show that, apart from some obvious exceptions, the number of trinomials vanishing at given complex numbers is bounded by an absolute constant. When the numbers are algebraic, we also bound effectively the degrees and the heights of these…

Until recently, the only known method of finding the roots of polynomials over prime power rings, other than fields, was brute force. One reason for this is the lack of a division algorithm, obstructing the use of greatest common divisors.…

While the separation (the minimal nonzero distance) between roots of a polynomial is a classical topic, its absolute counterpart (the minimal nonzero distance between their absolute values) does not seem to have been studied much. We…

Let $f \in { \mathbb R} ( t) [x]$ be given by $ f(t, x) = x^n + t \cdot g(x) $ and $\beta_1 < \dots < \beta_m$ the distinct real roots of the discriminant $\Delta_{(f, x)} (t)$ of $f(t, x)$ with respect to $x$. Let $\gamma$ be the number of…

This paper revisits an algorithm for isolating real roots of univariate polynomials based on continued fractions. It follows the work of Vincent, Uspen- sky, Collins and Akritas, Johnson and Krandick. We use some tricks, especially a new…

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 prove that any pair of bivariate trinomials has at most 5 isolated roots in the positive quadrant. The best previous upper bounds independent of the polynomial degrees counted only non-degenerate roots and even then gave much larger…

We construct parametric families of (monic) reducible polynomials having two roots very close to each other.

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a real coefficient polynomial. They can be approximated at a low computational cost if the…

We consider the problem of finding a sparse multiple of a polynomial. Given f in F[x] of degree d over a field F, and a desired sparsity t, our goal is to determine if there exists a multiple h in F[x] of f such that h has at most t…

Let f be a degree D univariate polynomial with real coefficients and exactly m monomial terms. We show that in the special case m=3 we can approximate within eps all the roots of f in the interval [0,R] using just O(log(D)log(Dlog(R/eps)))…

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…

For an odd prime $p$, we say a polynomial $f\in \mathbb F_p[X]$ computes square roots if $f(a)^2=a$ for all nonzero, perfect squares $a\in \mathbb F_p$. When $p\equiv 3 \mod 4$, it is easy to see that $f(X)=X^{\frac{p+1}{4}}$ is the…

Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…

For a commutative ring $R$, a polynomial $f\in R[x]$ is called separable if $R[x]/f$ is a separable $R$-algebra. We derive formulae for the number of separable polynomials when $R = \mathbb{Z}/n$, extending a result of L. Carlitz. For…