Contents - Student subdomain for University of Bath

4.4. FURTHER APPLICATIONS 139
2 7 3 3 a 1 + 2 7 3 2 a 2 + 2 7 3a 3 + 2 7 a 4 + 2 4 3 5 a 5 + a 6 = 9009
(
2 7 3 3) 2
a1 + ( 2 7 3 2) 2
a2 + ( 2 7 3 ) 2
a3 + ( 2 7) 2
a4 + ( 2 4 3 3) 2
a5 + a 6 = 28551425
(
2 7 3 3) 3
a1 + ( 2 7 3 2) 3
a2 + ( 2 7 3 ) 3
a3 + ( 2 7) 3
a4 + ( 2 4 3 3) 3
a5 + a 6 = 10 . . .
(
2 7 3 3) 4
a1 + ( 2 7 3 2) 4
a2 + ( 2 7 3 ) 4
a3 + ( 2 7) 4
a4 + ( 2 4 3 3) 4
a5 + a 6 = 37 . . .
(
2 7 3 3) 5
a1 + ( 2 7 3 2) 5
a2 + ( 2 7 3 ) 5
a3 + ( 2 7) 5
a4 + ( 2 4 3 3) 5
a5 + a 6 = 13 . . .
(
2 7 3 3) 6
a1 + ( 2 7 3 2) 6
a2 + ( 2 7 3 ) 6
a3 + ( 2 7) 6
a4 + ( 2 4 3 3) 6
a5 + a 6 = 51 . . .
(4.15)
This is indeed a system of linear equations
4.3.4 Conclusions
Let us look at the questions on page 114.
1. Are there evaluations which are not merely good in the sense of the previous
section, but also satisfy the Zippel assumption? The answer is yes.
For the evaluation y n = v to be good, v must not divide both leading
coefficients (checked for in random) and must not be a root of the relevant
resultant (Corollary 11).
2. How can we tell if an evaluation should be used?
3.
4.
5.
4.4 Further Applications
4.4.1 Matrix Determinants
The determinant det(M) of an n×n matrix M is simply a (rather large — a sum
of n! products each of n coefficients) polynomial in the entries of the matrix, and
polynomial arithmetic commutes with modular evaluation, and with replacing
variables by values (y = v evaluation). This gives the following unconditional
result
Proposition 45 det(M)| p = det(M| p ) and det(M)| y=v = det(M| y=v ).
Hence we have simple answer to the questions on page 114.
1. Are there good reductions from R?: yes, every reduction.
2. How can we tell if R i is good? — always.
3. How many reductions should we take? This depends on bounding the size
of det(M). In the polynomial case, this is easy from the polynomial form
of the determinant.