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Old 2004-03-09, 16:24   #19
R.D. Silverman
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Nov 2003

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Originally Posted by xilman
The root can be any size you like. What's important is the size of the coefficients of the polynomials.

Actually, that's an oversimplification, but it's a good first cut. If you want more details, dig out Brian Murphy's thesis.

An example may help. NFSNET is currently sieving the 201-digit composite cofactor of 10^223+1. We are using the polynomials x^6+10 and 1 - (10^37)*x which share a root 10^(-37) modulo 10^223+1. The latter value is a 201-digit number.

Actually Paul, the size of the root does matter. Let's use your example
and look at a 'typical' lattice point. (say (3x10^6, 3x10^6)) (choose
another if you like)

We have two polynomials. The corresponding norms at lattice point (b,a)
are a + 10^37 b and a^6 + 10b^6. For our typical point, the norms are
about 3x10^43 and 7x10^39. Note that the linear norm is larger. The
product is about 2x10^82

If we were to use a quintic, the linear norm becomes about 3 x 10^51
while the algebraic norm shrinks to 2 x 10^33. The product is now about
6 x 10^84, i.e. larger.

A septic would yield norms of about 3x10^38 and 2 x 10^46.

Having equal norms would be optimal.

The size of the root affects the norm of the linear polynomial. There is a
ying-yang effect. Reducing one norm increases the other and vice versa.
We want the product to be as small as possible averaged over the sieve

See my recent paper for a more detailed analysis.
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