Construction of irreducible polynomials through rational transformations

Panario, Daniel; Reis, Lucas; Wang, Qiang

doi:10.1016/j.jpaa.2019.106241

Cited by 8 publications

(12 citation statements)

References 13 publications

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“…Here, we have from (37), σ(x 4 ) = P 1 , σ((x + 1) 2a 2 ) = P 2 , and σ(R 2a 3 3 ) = P 3 . By (39) and (40) we have v R 3 (P 2 + 1) = 2 n 2 and v R 3 (P 3 + 1) = 1. Thus, v R 3 ((P 2 + 1)(P 3 + 1) = v R 3 (P 2 + 1) + v R 3 (P 3 + 1) = 2 n 2 + 1.…”

Section: Proof Of Theoremmentioning

confidence: 97%

Fixed points of the sum of divisors function on $F_2[x]$

Gallardo¹

2023

Preprint

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We work an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points F of the sum of divisors function σ : F 2 [x] → F 2 [x] (defined mutatis mutandi like the usual sum of divisors over the integers) of the form F := A 2 • S, S square-free, with ω(S) ≤ 3, coprime with A, for A even, of whatever degree, under some conditions. This gives a characterization of 5 of the 11 known fixed points of σ in F 2 [x]. so that σ(120) = 1+2+3+4+5+6+8+10+12+15+20+24+30+40+60+120 = 360. Already here we see that we can compute σ(120) more efficiently as follows: Since 120 = 2 3 • 3 • 5 and σ(x • y) = σ(x) • σ(y) provided that x, y has no common factors, we can compute:In a nutshell, in the present paper we study some arithmetic properties of an analogue to the function n → σ(n), in which we replace n by a polynomial A(x) with coefficients 0 and 1 only, and compute with 0, 1 as usual, besides the rule 1 + 1 = 0 that replaces the usual rule 1 + 1 = 2. The field F 2 = {0, 1} in which we compute the coefficients of A(x) is the simplest of all finite fields.For readers less familiar with finite fields, we recommend to look first at section 2 for a simple computation with binary polynomials. Then, to look at subsections 1.1, and 1.2 below. And, finally, come back to look at the rest of this Introduction.For all readers, we added some information about our choice of the finite field F 2 for the coefficients of our polynomials (see subsections 1.1, and 1.2) at the end of this Introduction. We also added a few comments about the role played by some small degree irreducible binary polynomials as prime factors of our perfect polynomials. This comes from an observation of one of the referees.The paper being a little technical, we hope the following considerations will be helpful for the reader.We now introduce some definitions and notation to explain the original arithmetic problem over the integers that motivated the study of our variant over the binary polynomials in F 2 [x], and the link between them as well.Let A ∈ F 2 [x] be an irreducible polynomial, then we say that A is prime. A polynomial M ∈ F 2 [x] is Mersenne (an analogue of a Mersenne number: 2 n − 1) if M + 1 is a product of powers of x and powers of x + 1. We say that M + 1 splits. When a Mersenne polynomial M is irreducible, we say that M is a Mersenne prime. Given a binary polynomial B, a binary polynomial A in the sub-ring, if all coefficients of A are equal to 1; when B = x, we say simply that A is complete. A binary polynomial B is odd if B(0) = B(1) = 1, otherwise B is even. More standard notation follows. We let ω(P ) denote the number of pairwise distinct prime factors of P ∈ F q [x]. Likewise, we let v P (A) denote the valuation of the prime P in the binary polynomial A, i.e., the least positive integer m, such that P m | A but P m+1 ∤ A, we also write this as P m ||A. Finally, we let F 2 denote a fixed algebraic closure of F 2 .We recall that a binary perfect polynomial A (see [11,14,16,19,26,29,31,32,33]) is defined by the ...

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Section: Proof Of Theoremmentioning

confidence: 97%

Fixed points of the sum of divisors function on $F_2[x]$

Gallardo¹

2023

Preprint

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“…What properties have the composite polynomial F (x) = g 2 (x) deg(f (x)) f (g(x))? (see [1,2,3,5,6,8,9,19,20,21,23,24,25]). More generally, g(x) has been chosen as a power of x, as a linearized polynomial, e.g., g(x) = x p r − x, or as an appropriate quotient A/B of two other polynomials.…”

Section: Introductionmentioning

confidence: 99%

“…More generally, g(x) has been chosen as a power of x, as a linearized polynomial, e.g., g(x) = x p r − x, or as an appropriate quotient A/B of two other polynomials. In particular, recently Panario et al [23] worked on the case g(x) = 1/(cx + 1) = 1, in order to obtain conditions such that f (x) irreducible implies F (x) irreducible. More generally, one finds in Lidl and Niederreiter [22], and in Swan [27], the most basic results about polynomials over finite fields, including results about the factorization of these polynomials.…”

Section: Introductionmentioning

confidence: 99%

On the equation Π_P Φ_3(P) = Π_P P^2 over F_2[x]

Gallardo

2022

GJOM

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“…Technically, observe that the following problem has attracted some interest (see [1,2,3,9,10,12,13,35,36,38,39,40,41]). Given an irreducible polynomial f over a finite field F q , given a polynomial g(x) over the same field.…”

Section: Introductionmentioning

confidence: 99%

Fixed points of the sum of divisors function on ${{\mathbb{F}}}_2[x]$

Gallardo¹

2022

Glas. Mat. Ser. III

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We work on an analogue of a classical arithmetic problem over polynomials. More precisely, we study the fixed points $F$ of the sum of divisors function $\sigma : {\mathbb{F}}_2[x] \mapsto {\mathbb{F}}_2[x]$ (defined mutatis mutandi like the usual sum of divisors over the integers) of the form $F := A^2 \cdot S$, $S$ square-free, with $\omega(S) \leq 3$, coprime with $A$, for $A$ even, of whatever degree, under some conditions. This gives a characterization of $5$ of the $11$ known fixed points of $\sigma$ in ${\mathbb{F}}_2[x]$.

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Construction of irreducible polynomials through rational transformations

Cited by 8 publications

References 13 publications

Fixed points of the sum of divisors function on $F_2[x]$

Fixed points of the sum of divisors function on $F_2[x]$

On the equation Π_P Φ_3(P) = Π_P P^2 over F_2[x]

Fixed points of the sum of divisors function on \({{\mathbb{F}}}_2[x]\)

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