Date: Thu, 27 Apr 95 01:18:36 PDT
From: robert@cs.caltech.edu (Rob actually)
To: rusin@math.niu.edu
Subject: Re: Q: Series involving sech

>Lenstra-Lenstra-Lovasz

Yes.  For instance in pari/gp (with default precision 28 digits):

------------------------------------------------------------------------------
? x=sqrt(2)+sqrt(3)
%1 = 3.146264369941972342329135065
? k=10^20;m=[1,0,0,0,0;0,1,0,0,0;0,0,1,0,0;0,0,0,1,0;0,0,0,0,1;k,round(k*x),round(k*x^2),round(k*x^3),round(k*x^4)]
%2 =
[1 0 0 0 0]

[0 1 0 0 0]

[0 0 1 0 0]

[0 0 0 1 0]

[0 0 0 0 1]

[100000000000000000000 314626436994197234233 989897948556635619639 3114480645422394117857 9798979485566356196395]

? ?lllkerim
lllkerim(x)=kernel and lll reduction of the vectors forming the integral
matrix x.
? lllkerim(m)[2]
%3 = 
[1 29548 69913 80758 -186770]

[0 -124586 -29481 -37402 -95123]

[-10 -16827 40483 56619 44179]

[0 -51059 110039 -89404 54557]

[1 21627 -38831 23073 -16843]

------------------------------------------------------------------------------

The first row is a short vector, in fact 1-10*x^2+x^4 = 0.

A few places that this is described include lots of papers by the
Lenstras, Henri Cohen's book "A Course in Computational Algebraic
Number Theory", Zippel's "Effective Polynomial Computation" (full of
errors...)

Bye,
  Rob.