Squarefree Values of Multivariable Polynomials

Squarefree Values of Polynomials

where m is any integer > n:We will be interested in estimating the number of polynomials f(x) for which there exists an integer m such that f(m) is squarefree. This property should hold for all polynomials f(x) for which Nf is squarefree. However, this seems to be very di cult to establish. Nagel [8] showed that if f(x) 2 Z[x] is an irreducible quadratic and Nf is squarefree, then f(m) is squar...

Squarefree Values of Polynomials All

A construction of polynomials with squarefree discriminants

For any integer n ≥ 2 and any nonnegative integers r, s with r + 2s = n, we give an unconditional construction of infinitely many monic irreducible polynomials of degree n with integer coefficients having squarefree discriminant and exactly r real roots. These give rise to number fields of degree n, signature (r, s), Galois group Sn, and squarefree discriminant; we may also force the discrimina...

Analyzing Multiplicities of a Zero-dimensional Regular Set's Zeros Using Pseudo Squarefree Decomposition

In this paper, we are concerned with the problem of counting the multiplicities of a zero-dimensional regular set’s zeros. We generalize the squarefree decomposition of univariate polynomials to the so-called pseudo squarefree decomposition of multivariate polynomials, and then propose an algorithm for decomposing a regular set into a finite number of simple sets. From the output of this algori...

Squarefree Decomposition of Univariate Polynomials Depending on a Parameter. Application to the Integration of Parametric Rational Functions

The squarefree decomposition of a univariate polynomial, P (y) ∈ K[y], K being a characteristic zero field, is one of the most basic problems in computer algebra (after the greatest common divisor computation, probably the most basic one). The parametric case has not usually been considered in the literature and this paper is dedicated to showing how the case of a single parameter can easily be...