In mathematics, a square-free polynomial is a univariate polynomial (over a field or an integral domain) that has no multiple root in an algebraically...

for computing the square-free part of an integer, or even for determining whether an integer is square-free. In contrast, polynomial-time algorithms are...

roots of a polynomial are the roots of the GCD of the polynomial and its derivative, and further GCD computations allow computing the square-free factorization...

modulo p. Algorithm: SFF (Square-Free Factorization) Input: A monic polynomial f in Fq[x] where q = pm Output: Square-free factorization of f R ← 1 #...

distinct roots is equal to the degree of the polynomial. This concept is closely related to square-free polynomial. If K is a perfect field then the two concepts...

polynomials, other expressions, or values in systems of mathematical values other than the numbers. For instance, the square of the linear polynomial...

replacing "irreducible polynomial" by "square-free polynomial" in the description of the outcome. This allows replacing polynomial factorization by the...

especially in the field of algebra, a polynomial ring or polynomial algebra is a ring formed from the set of polynomials in one or more indeterminates (traditionally...

it divisible by a square number). Common examples of square-free elements include square-free integers and square-free polynomials. Prime number David...

real roots of a polynomial Root-finding of polynomials – Algorithms for finding zeros of polynomials Square-free polynomial – Polynomial with no repeated...

{\displaystyle \int {\frac {x^{k}\,dx}{\sqrt {Q(x)}}}} where Q is a square-free polynomial of any given degree > 4. The allowable power k has to be determined...

resultant of two polynomials is a polynomial expression of their coefficients that is equal to zero if and only if the polynomials have a common root...

mathematics and computer algebra, factorization of polynomials or polynomial factorization expresses a polynomial with coefficients in a given field or in the...

called square-free factorization, is based on the multiple roots of a polynomial being the roots of the greatest common divisor of the polynomial and its...

with polynomials with integer coefficients, and intervals ending with rational numbers. Also, the polynomials are always supposed to be square free. There...

coefficient for a polynomial of even degree, and the opposite sign for a polynomial of odd degree. In the case of a non-square-free polynomial, if neither a...

In mathematics, a polynomial is a mathematical expression consisting of indeterminates (also called variables) and coefficients, that involves only the...

quadratic polynomial a x 2 + b x + c {\displaystyle ax^{2}+bx+c} is b 2 − 4 a c , {\displaystyle b^{2}-4ac,} the quantity which appears under the square root...

In mathematics, a homogeneous polynomial, sometimes called quantic in older texts, is a polynomial whose nonzero terms all have the same degree. For example...

example, the fundamental theorem of algebra asserts that a univariate square-free polynomial of degree n has exactly n complex roots. In the multivariate case...

In mathematics, the nth cyclotomic polynomial, for any positive integer n, is the unique irreducible polynomial with integer coefficients that is a divisor...

corollary of this theorem is that, if two irreducible polynomials (or more generally two square-free polynomials) define the same hypersurface, then one is the...

theorem of arithmetic Square-free Square-free integer Square-free polynomial Square number Power of two Integer-valued polynomial Rational number Unit...

In mathematics, a positive polynomial (respectively non-negative polynomial) on a particular set is a polynomial whose values are positive (respectively...

In mathematics, the degree of a polynomial is the highest of the degrees of the polynomial's monomials (individual terms) with non-zero coefficients. The...

mathematics, a form (i.e. a homogeneous polynomial) h(x) of degree 2m in the real n-dimensional vector x is sum of squares of forms (SOS) if and only if there...

f {\displaystyle V_{f}} (in particular, for every non-constant square-free polynomial f {\displaystyle f} of two variables, the case of plane curves)...

of mathematics, the minimal polynomial of an element α of an extension field of a field is, roughly speaking, the polynomial of lowest degree having coefficients...

P(y)} is a square-free polynomial, that is that the solutions of P ( y ) = 0 {\displaystyle P(y)=0} are all different. Indeed, the square-free factorization...

1979). "Deterministic CFL's are accepted simultaneously in polynomial time and log squared space". Proceedings of the eleventh annual ACM Symposium on...