Re: reversing a series modulo

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Sat, May 03, 2025 at 02:38:33PM -0400, Max Alekseyev wrote:
> Hello,
> 
> Reversing a series and taking it modulo 5 works, but not the opposite way
> around. Why?
> 
> ? f = x + x^2 + 4*x^6 + 4*x^7 + x^11 + x^12 + 4*x^16 + 4*x^17 + O(x^21)
> %1 = x + x^2 + 4*x^6 + 4*x^7 + x^11 + x^12 + 4*x^16 + 4*x^17 + O(x^21)
> ? serreverse(f) * Mod(1,5)
> %2 = Mod(1, 5)*x + Mod(4, 5)*x^2 + Mod(2, 5)*x^3 + Mod(4, 5)*x^5 + Mod(4,
> 5)*x^6 + Mod(2, 5)*x^8 + Mod(3, 5)*x^10 + Mod(3, 5)*x^11 + Mod(4, 5)*x^12 +
> Mod(4, 5)*x^17 + Mod(3, 5)*x^18 + O(x^21)
> ? serreverse(f * Mod(1,5))
>   ***   at top-level: serreverse(f*Mod(1,5))
>   ***                 ^----------------------
>   *** serreverse: impossible inverse in Fl_inv: Mod(0, 5).

The formula used by PARI is
g=intformal(1/subst(f',x,g))
which involves division by small primes (when integrating)

What you can do is 

? serreverse(f*(1+O(5^2)))*Mod(1,5)
%34 = Mod(1, 5)*x + Mod(4, 5)*x^2 + Mod(2, 5)*x^3 + Mod(4, 5)*x^5 + Mod(4, 5)*x^6 + Mod(2, 5)*x^8 + Mod(3, 5)*x^10 + Mod(3, 5)*x^11 + Mod(4, 5)*x^12 + Mod(4, 5)*x^17 + Mod(3, 5)*x^18 + O(x^21)

Cheers,
Bill