It is well-known that a real number is a rational if, and only, if its decimal expansion is periodic (see Wikipedia). This result is a finiteness result, as it says the decimal expansion is, in a way, not really infinite, as it consists in the repeating infinitely a finite block.
A similar finiteness result exists for Laurent series , which are the analogues of reals modulo 1. Namely, associate to
the infinite matrix
. Then Hankel’s theorem states that
is rational if and only if the matrix is of finite rank.
We show here a quantitative version of this statement. Let
The rank of this matrix is related to Diophantine properties of .
To see this, we introduce for the fractional part
and the norm
, where
is the largest
such that
. Then we have the following result.
If the rank of is at most
, then there exist a decomposition
and a monic polynomial
of degree
such that
. The converse holds too.
The classical theory of Hankel matrices (see Chapter X, Paragraph 10 of Gantmacher’s book) tells us that if the rank of the matrix is , then there exist integers
and
such that
and the
first rows are linearly independent, each of the following
rows is a linear combination of them,
and the minor formed of the first and last
rows and columns is nonzero. Let
be the rows of
. Write
where
is the last coefficient of
and
the row of the first
coefficients. By the above, we have a relation
. In fact, we shall show that for any
, we have
This equation holds for . So we argue by induction and assume it holds for any
for some
and prove it for
. To start with, the displayed equation implies that
Applying the induction hypothesis iteratively, we find coefficients for
and
such that
and
These coefficients satisfy the initial condition and the recurrence relations
for
and
.
On the other hand, we know that there exist coefficients
such that
Comparing the last equation to the one involving , and using the linear independence of the first
lines, we find that
.
Thus . But
by definition of the coefficients
. We infer that
,
which conclude the inductive argument.
To see the connexion with Diophantine properties of , notice that
is the row of the first
coefficients of
. Thus the validity of the identity for
for all
implies that
where satisfies
.
That is, using the polynomial , we find that
which concludes the proof of the direct statement. The converse is straightforward.
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