Download Mathematical Computations Using Bergman

3.2. MONOMIALS AND POLYNOMIALS IN BERGMAN
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Internally, domain polynomials are represented in two ways, as reductand
polynomials (redands) or as reductor polynomials (redors). The difference is
not as the difference between “and” and “or”, but rather as the difference
between “operand” and “operator” or may be better to say as a difference
between “summand” and “multiplicator”. Redands have redand coefficients
and redors have redor coefficients. What is the difference? Redands are
what we expect from the coefficient representation, but redors have form,
more convenient for the multiplication.
Let us consider first an example. Suppose that our main field is F5 and
we consider a representation of the polynomial x2 + 3xy + 2y 2.
In the redand form it will be represented as a list
(P (1 hx2 i) (3 hxyi) (2 hy 2 i)),
where P stands for a pointer which we do not discuss now, hx2 i stands for
representation of the pure monomial x2 as it was discussed above. But just
now we are interested in the coefficients. 1, 3, 2 are exactly what we expect.
What we do not expect from the very beginning is that in the redor form the
same polynomial will be written as
(P (0 hx2 i) (3 hxyi) (1 hy 2 i)).
What does it mean? Recall that 2 is a primitive element in F5 , so
20 = 1, 21 = 2, 22 = 4, 23 = 3
and all non-zero elements in F5 can be represented by the corresponding
logarithms. So, we can represent our elements in two different ways according
to the table:
Normal representation
Logarithmic representation
0
NIL
NIL
1 2
1 2
0 1
3 4
3 4
3 2
and it is quite clear now that redor polynomials use logarithmic representation in this case.
More exactly redors coefficients are written in the form that is more convenient for the multiplication in the corresponding domain. In reality it differs
now from the redand coefficients only in the case when we work in odd prime
characteristics with the logarithmic tables. So practically the user can skip