On the Degree of Univariate Polynomials Over the Integers

Rocky Mountain Journal of Mathematics, 2007

Let m ≥ 2 and k ≥ 2 be integers and let R be a commutative ring with a unit element denoted by 1. A k-th power diophantine m-tuple in R is an m-tuple (a 1 , a 2 , . . . , a m ) of non-zero elements of R such that a i a j + 1 is a k-th power of an element of R for 1 ≤ i < j ≤ m. In this paper, we investigate the case when k ≥ 3 and R = K[X], the ring of polynomials with coefficients in a field K of characteristic zero. We prove the following upper bounds on m, the size of diophantine m-tuple: m ≤ 5 if k = 3; m ≤ 4 if k = 4; m ≤ 3 for k ≥ 5; m ≤ 2 for k even and k ≥ 8.

2010

Third International Meeting on Integer-Valued Polynomials and Problems in Commutative Algebra Combinatorial, arithmetical, algebraic, topological and dynamical aspects CIRM International center of mathematics meetings, 163 avenue de luminy 13288 MARSEILLE Organizing committee Sabine EVRARD AMIENS Youssef FARES AMIENS Amandine LERICHE AMIENS Jean-Luc CHABERT AMIENS Paul-Jean CAHEN MARSEILLE Scientific committee Paul-Jean CAHEN FRANCE Jean-Luc CHABERT FRANCE Stefania GABELLI ITALY Byung KANG SOUTH COREA Roger WIEGAND USA MONDAY THUESDAY WEDNESDAY THURSDAY FRIDAY 9H 9H/9H30 9H/9H30 9H40/10H 9H40/10H 9H50/10H30 9H50/10H30 Alice FABBRI Valentina BARUCCI 10H 10H10/10H30 10H10/10H30 Said El BAGHDADI Marco FONTANA 11H 11H/11H40 11H/11H30 11H/11H30 11H/11H30 11H/11H30 11H40/12H 11H40/12H10 11H40/12H 11H40/12H 11H50/12H30 Vadim PONOMARENKO Gabriele FUSACCHIA Gabriel PICAVET 12H 12H10/12H30 12H10/12H30 12H10/12H30 Faten KHOUJA Driss KARIM Amor HAOUAOUI 14H/14H20 Mohamed KHALIFA 14H30 14H30/15H...

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.

2003

Given r&gt; 2, we establish a good upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer co- ecients mapping the \cube&quot; with real coordinates from ( r; r )i nto ( t;t). This directly translates to a nice statement in logic (more specically recursion theory) with a correspond- ing phase transition case of 2 being open. We think this situation will be of real interest to logicians. Other related questions are also considered. In most of these problems our main idea is to write the multivariate polynomials as a linear combination of products of scaled Chebyshev polynomials of one variable. In some private communications, Harvey Friedman raised the following problem: given r&gt; 2, give an upper bound for the number of multivariate polynomials (with as many variables and with as large degree as we wish) with integer coecients mapping the \cube&quot; with real variables from ( r; r )i nto ( t; t). Robin Peman...

Journal of Algebra, 2011

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.

Functiones et Approximatio Commentarii Mathematici

In this paper, we study the derivatives of an integer-valued polynomial of a given degree. Denoting by E n the set of the integer-valued polynomials with degree ≤ n, we show that the smallest positive integer c n satisfying the property: ∀P ∈ E n , c n P ′ ∈ E n is c n = lcm(1, 2,. .. , n). As an application, we deduce an easy proof of the well-known inequality lcm(1, 2,. .. , n) ≥ 2 n−1 (∀n ≥ 1). In the second part of the paper, we generalize our result for the derivative of a given order k and then we give two divisibility properties for the obtained numbers c n,k (generalizing the c n 's). Leaning on this study, we conclude the paper by determining, for a given natural number n, the smallest positive integer λ n satisfying the property: ∀P ∈ E n , ∀k ∈ N: λ n P (k) ∈ E n. In particular, we show that: λ n = p prime p ⌊ n p ⌋ (∀n ∈ N).

International Journal of Algebra, 2015

Consider an algebraic number field K of degree n, A K is its ring of integers and a prime number p inert in K. Let F (u 1 ,. .. , u n , x) be the generic polynomial of integers of K. We will study in advance the stability of this polynomial and then, we will apply it in order to obtain all the monic irreducible polynomials in F p [x] of degree d dividing n.

Experimental Mathematics, 2000

In this paper, we prove that there does not exist a set of 8 polynomials (not all constant) with coefficients in an algebraically closed field of characteristic 0 with the property that the product of any two of its distinct elements plus 1 is a perfect square.