A367643 Number of equivalence classes of degree n integer polynomials whose discriminants are powers of 2 (in absolute value).

1, 4, 4, 39, 51, 135

Offset: 1

Author

Robin Visser, Nov 25 2023

Keywords

• nonn
• hard
• more

Comments

Here, two degree n integer polynomials f(x) and g(x) are considered equivalent if there exist integers a, b, c, d such that a*d - b*c is not zero and (cx+d)^n * f((ax+b)/(cx+d)) is some nonzero rational multiple of g(x).

Examples

			For n = 1, every linear (degree 1) polynomial is equivalent to x and has discriminant 1, so a(1) = 1.
For n = 2, the a(2) = 4 equivalence classes are represented by the degree 2 polynomials x^2 + x, x^2 + 1, x^2 + 2, and x^2 - 2.  These have discriminants 1, -4, -8, and 8 respectively.
For n = 3, the a(3) = 4 equivalence classes are represented by the degree 3 polynomials x^3 + x, x^3 - x, x^3 + 2*x, and x^3 - 2*x.  These have discriminants -4, 4, -32, and 32 respectively.
		

Links

• J.-H. Evertse and K. Gyory, Discriminant equations in Diophantine number theory, Cambridge University Press, Cambridge, 2017. xviii+457 pp.
• J. R. Merriman and N. P. Smart, Curves of genus 2 with good reduction away from 2 with a rational Weierstrass point, Math. Proc. Cambridge Philos. Soc. 114 (1993), no. 2, 203-214.
• N. P. Smart, S-unit equations, binary forms and curves of genus 2, Proc. London Math. Soc. (3) 75 (1997), no. 2, 271-307.