Irreducibility of polynomials modulo p via Newton polytopes

Turkish Journal of Mathematics, 2011

Motivated by the Dubickas's result in [1], which computes the probability of the irreducible polynomials by Eisenstein's criterion for some families of polynomials in [x], we calculate the probabilities which represent the ratio of absolutely irreducible multivariate polynomials by the polytope method in some families of polynomials over arbitrary fields.

Journal of Symbolic Computation, 2012

We present algorithms revealing new families of polynomials allowing sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we show that the case of honest n-variate (n + 1)-nomials is doable in NP and, for p exceeding the Newton polytope volume and not dividing any coefficient, in constant time. Furthermore, using the theory of linear forms in p-adic logarithms, we prove that the case of trinomials in one variable can be done in NP. The best previous complexity bounds for these problems were EXPTIME or worse. Finally, we prove that detecting p-adic rational roots for sparse polynomials in one variable is NP-hard with respect to randomized reductions. The last proof makes use of an efficient construction of primes in certain arithmetic progressions. The smallest n where detecting p-adic rational roots for n-variate sparse polynomials is NP-hard appears to have been unknown.

Journal of Symbolic Computation, 2012

We present algorithms revealing new families of polynomials admitting sub-exponential detection of p-adic rational roots, relative to the sparse encoding. For instance, we prove NPcompleteness for the case of honest n-variate (n + 1)-nomials and, for certain special cases with p exceeding the Newton polytope volume, constant-time complexity. Furthermore, using the theory of linear forms in p-adic logarithms, we prove that the case of trinomials in one variable can be done in NP. The best previous complexity upper bounds for all these problems were EXPTIME or worse. Finally, we prove that detecting p-adic rational roots for sparse polynomials in one variable is NP-hard with respect to randomized reductions. The last proof makes use of an efficient construction of primes in certain arithmetic progressions. The smallest n where detecting p-adic rational roots for n-variate sparse polynomials is NP-hard appears to have been unknown.

Journal of Number Theory, Vol. 147, 549-589, 2015

Let $A$ be a Dedekind domain whose field of fractions $K$ is a global field. Let $p$ be a non-zero prime ideal of $A$, and $K_p$ the completion of $K$ at $p$. The Montes algorithm factorizes a monic irreducible separable polynomial $f(x)\in A[x]$ over $K_p$, and it provides essential arithmetic information about the finite extensions of $K_p$ determined by the different irreducible factors. In particular, it can be used to compute a $p$-integral basis of the extension of $K$ determined by $f(x)$. In this paper we present a new and faster method to compute $p$-integral bases, based on the use of the quotients of certain divisions with remainder of $f(x)$ that occur along the flow of the Montes algorithm.

2002

For F ∈ Z[X], let Ψ F (x, y) denote the number of positive integers n not exceeding x such that F (n) is free of prime factors > y. Our main purpose is to obtain lower bounds of the form Ψ F (x, y) x for arbitrary F and for y equal to a suitable power of x. Our proofs rest on some results and methods of two articles by the third author concerning localization of divisors of polynomial values. Analogous results for the polynomial values at prime arguments are also obtained.

We obtain an estimate on the average cardinality of the value set of any family of monic polynomials of F q [T ] of degree d for which s consecutive coefficients a d−1 ,. .. , a d−s are fixed. Our estimate holds without restrictions on the characteristic of F q and asserts that V(d, s, a) = µ d q + O(1), where V(d, s, a) is such an average cardinality, µ d := d r=1 (−1) r−1 /r! and a := (a d−1 ,. .. , d d−s). We provide an explicit upper bound for the constant underlying the O-notation in terms of d and s with "good" behavior. Our approach reduces the question to estimate the number of F q-rational points with pairwise-distinct coordinates of a certain family of complete intersections defined over F q. We show that the polynomials defining such complete intersections are invariant under the action of the symmetric group of permutations of the coordinates. This allows us to obtain critical information concerning the singular locus of the varieties under consideration, from which a suitable estimate on the number of F q-rational points is established.

arXiv (Cornell University), 2022

Using an explicit version of Selberg's upper sieve, we obtain explicit upper bounds for the number of n ≤ x such that a non-empty set of irreducible polynomials F i (n) with integer coefficients are simultaneously prime; this set can contain as many polynomials as desired. To demonstrate, we present computations for some irreducible polynomials and obtain an explicit upper bound for the number of Sophie Germain primes up to x, which have practical applications in cryptography. 2. Selberg's upper sieve In this section, we introduce Selberg's upper sieve from [6] and make every aspect of this sieve explicit. That is, all implied constants will be described using constants. 2.1. Notation and initial result. Suppose A = (a n) is a finite sequence of non-negative real numbers a n , P is a set of primes, and P C is the complement of P in the set of all primes. The objective of sieve theory is to obtain bounds for the sifted set S(A, P, z) = #{a ∈ A : (a, P (z)) = 1} such that P (z) = p∈P p≤z p.

Finite Fields and Their Applications, 2012

We show that, for any integer ℓ with q − √ p − 1 ≤ ℓ < q − 3 where q = p n and p > 9, there exists a multiset M satisfying that 0 ∈ M has the highest multiplicity ℓ and b∈M b = 0 such that every polynomial over finite fields Fq with the prescribed range M has degree greater than ℓ. This implies that Conjecture 5.1. in [1] is false over finite field Fq for p > 9 and k := q − ℓ − 1 ≥ 3.

2017

In this paper, assuming a conjecture of Vojta&#39;s on bounded degree algebraic numbers and a quantitative version of Northcott&#39;s theorem over number field $k$, we show the existence of explicit lower and upper bounds for the number of polynomials $f\in k[x]$ of degree $r$ whose irreducible factors have multiplicity strictly less than $s$ and moreover $f(b_1),\cdots, f(b_M)$ are $s$-powerful values for a certain integer $M$, where $b_i$&#39;s belong to a sequence of the pairwise distinct element of $k$ that satisfy certain conditions. Our results improve the recent work of H. Pasten on the subject.

Finite Fields and Their Applications, 2008

Following Beard, we call a polynomial over a finite field F q perfect if it coincides with the sum of its monic divisors. The study of perfect polynomials was initiated in 1941 by Carlitz's doctoral student Canaday in the case q = 2, who proposed the still unresolved conjecture that every perfect polynomial over F 2 has a root in F 2 . Beard, et al. later proposed the analogous hypothesis for all finite fields. Counterexamples to this general conjecture were found by Link (in the cases q = 11, 17) and Gallardo & Rahavandrainy (in the case q = 4). Here we show that the Beard-O'Connell-West conjecture fails in all cases except possibly when q is prime. When q = p is prime, utilizing a construction of Link we exhibit a counterexample whenever p ≡ 11 or 17 (mod 24). On the basis of a polynomial analog of Schinzel's Hypothesis H, we argue that if there is a single perfect polynomial over the finite field F q with no linear factor, then there are infinitely many. Lastly, we prove without any hypothesis that there are infinitely many perfect polynomials over F 11 with no linear factor.