Re: math -- values of f(x) (mod p)
- From: Olivier <Olve@xxxxxxxxxx>
- Date: Fri, 11 Apr 2008 09:38:55 +0200
quasi a écrit :
Two conjectures ...
Conjecture (1):
If n is an odd positive integer, then for all sufficiently large
primes p (depending on n), there does not exist f in Z_p[x], with
deg(f) = n, such that for all r in Z_p, f(r) is a square in Z_p.
Well, look at the curve y^2 = f(x). Weil's bound (Riemann hypothesis
on finite curves) tells you that, when f is not a square and has m
distinct roots, we have | sum_{x mod p} (f(x)/p) | <= (m-1) sqrt(q).
What about the distinct roots condition ? If I'm not mistaken, this
condition is to be understood in an algebraic closure, i.e. f and
f' coprime, but a confirmation would be welcomed :-)
If f(x) was a square for all x, then | sum_{x mod p} (f(x)/p) | = p
and that would violate the above bound when p is large enough with
respect to m.
HTH,
O.
.
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