Arithmetic of function fields

Number-theoretic analogues over F_q[T] and finite fields: polynomial primes, h-free polynomials, function-field adeles, positive-characteristic zeta values.

• Adeles of function fields
• Divisor problems for B-free polynomials
• Higher a-numbers in ℤ_p-towers
• Ihara constant
• Multiple zeta values in positive characteristic
• Prime factors of given multiplicity in h-free polynomials
• Twin primes and Chowla's conjecture over function fields

TheoremOdd perfect polynomials over \(\mathbb{F}_2\) are squares (Canaday)#✓fully proved

Let \(A\) be a polynomial over \(\mathbb{F}_2\) that is coprime to both \(X\) and \(X + 1\) (that is, odd: \(A\) has no linear factor) and that equals the sum of its monic divisors (that is, perfect). Then \(A\) is a square: \(A = B^2\) for some polynomial \(B\) over \(\mathbb{F}_2\).

Since every nonzero polynomial over \(\mathbb{F}_2\) is monic, the sum over monic divisors is the full divisor sum.

theorem odd_perfect_polynomial_is_square
    (A : Polynomial (ZMod 2))
    (hodd : IsCoprime A Polynomial.X ∧ IsCoprime A (Polynomial.X + 1))
    (hperfect : (finsum fun d : {d : Polynomial (ZMod 2) // d.Monic ∧ d ∣ A} =>
      (d : Polynomial (ZMod 2))) = A) :
    IsSquare A

Sources

• Luis H. Gallardo and Olivier Rahavandrainy, "A Polynomial Variant of Perfect Numbers," Journal of Integer Sequences 23 (2020), Article 20.8.6, Lemma oddperfect, lines 155–158 (odd and perfect defined lines 96–108), https://cs.uwaterloo.ca/journals/JIS/VOL23/Gallardo/gallardo7.tex attributed there to Canaday.

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