منابع مشابه
Reducibility of Polynomials
Reducibility without qualification means in this lecture reducibility over the rational field. Questions on such reducibility occupy an intermediate place between questions on reducibility of polynomials over an algebraically closed field and those on primality. I shall refer to these two cases as to the algebraic and the arithmetic one and I shall try to exhibit some of the analogies consideri...
متن کاملFactoring bivariate sparse (lacunary) polynomials
We present a deterministic algorithm for computing all irreducible factors of degree ≤ d of a given bivariate polynomial f ∈ K[x, y] over an algebraic number field K and their multiplicities, whose running time is polynomial in the bit length of the sparse encoding of the input and in d . Moreover, we show that the factors over Q of degree ≤ d which are not binomials can also be computed in tim...
متن کاملDivisibility Test for Lacunary Polynomials
Given two lacunary (i.e. sparsely-represented) polynomials with integer coefficients, we consider the decision problem of determining whether one polynomial divides the other. In the manner of Plaisted [6], we call this problem 2SparsePolyDivis. More than twenty years ago, Plaisted identified as an open problem the question of whether 2SparsePolyDivis is in P [7]. Some progress has been made si...
متن کاملInterpolation of Shifted-Lacunary Polynomials [Extended Abstract]
Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 < e2 < · · · < et, and the coefficients c1, . . . , ct ∈ Q \ {0} such that f (x) = c1(x − α)1 + c2(...
متن کاملIrreducibility testing of lacunary 0, 1-polynomials
A reciprocal polynomial g(x) ∈ Z[x] is such that g(0) 6= 0 and if g(α) = 0 then g(1/α) = 0. The non-reciprocal part of a monic polynomial f(x) ∈ Z[x] is f(x) divided by the product of its irreducible monic reciprocal factors (to their multiplicity). This paper presents an algorithm for testing the irreducibility of the nonreciprocal part of a 0, 1-polynomial (a polynomial having each coefficien...
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ژورنال
عنوان ژورنال: Acta Arithmetica
سال: 1970
ISSN: 0065-1036,1730-6264
DOI: 10.4064/aa-16-4-371-392