Re: conditions for this polynomial to be in Q[X]

In article <1160430576.041008.97130@xxxxxxxxxxxxxxxxxxxxxxxxxxx>,
<ben_r_jackson@xxxxxxxxxxx> wrote:

I'd like to get conditions for when
F(X) = a1*(X-b1)^n+...+ak*(X-bk)^n is a rational polynomial.

If you expand this out, you'll get some binomial coefficients, so
let's suppose your F equals \sum { \binomial n j } r_j X^{n-j}.
You want to describe the conditions on the a_i and b_i so that
each of the numbers r_j is rational, where
\sum_i a_i (b_i)^j = r_j for all j=0,1,...,n
Note that my naming of the coefficients implies that the equations
for one value of n are among the equations for larger values of n
(as long as k is fixed).

I treated this as a system of 2k polynomial equations in 2k unknowns
a_i and b_i, setting r_0, ..., r_{2k-1} to random rational values;
using Magma I computed Groebner bases for these ideals, and found in
every experiment that the equations were equivalent to a set of the form
a_1 = f_1, a_2 = f_2, ..., a_k = f_k,
b_1 + g_1 = 0,
(b_2)^2 + g_2 = 0,
...
(b_k)^k + g_k = 0
where the f_i and g_i are polynomials in b_2, b_3, ..., b_k.
(They are at most linear in b_2, quadratic in b_3, etc.
Also, g_i involves only the variables b_i, b_{i+1}, ...)

The interpretation of this calculation is as follows: what appears to be
true is that for almost all choices of rational numbers r_0, ..., r_{2k-1},
there are exactly k! possible values for the 2k-tuple (a_1, ..., b_k).
The k possible values of b_k are the roots of a polynomial of degree k
(whose coefficients are determined by the r_i); then for each of these
values of b_k we compute a polynomial whose roots are the k-1 possible
values of b_{k-1}; and so on back to solving a quadratic to get b_2,
and then reading off the values of b_1 and the a_i.

(With the a_i and b_i then all known, it is not surprising that the
values of the remaining r_i can be determined. That is, we have only
enough "degrees of freedom" to pick r_0 through r_{2k-1} arbitrarily.
Perhaps somewhat surprisingly, the other r_i can be expressed in
terms of r_0 through r_{2k-1} only, that is, the choices of roots
for the b_i make no difference in the value of r_{2k}, r_{2k+1}, ...)

Because of this "triangular" form of the equations, we conclude that
for n >= 2k-1, the a_i and b_i are algebraic, and indeed lie in
an extension of degree at most k! . Exactly which such extension contains
the a_i and b_i depends on the choice of r_0 through r_{2k-1}.

So you have what you want here, at least for generic choices of r_0 through
r_{2k-1}:

Ideally, for n sufficiently large with respect to k, I'd like to show
that the ai's and bi's are algebraic, must lie in number fields whose
degree depends on k in some way,...

dave

.

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