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 \alpha=\sum_{i=1}^{\infty}\alpha_i T^{-i} \in K((1/T)), which are the analogues of reals modulo 1. Namely, associate to \alpha the infinite matrix (\alpha_{(i+j)-1})_{i,j\in\mathbb{N}^2}. Then Hankel’s theorem states that \alpha is rational if and only if the matrix is of finite rank.

We show here a quantitative version of this statement. Let

\displaystyle{M_{\alpha,n}=\begin{pmatrix} \alpha_{1} & \cdots & \alpha_{n} & \\ \alpha_{2} & \cdots & \alpha_{n+1} & \\ \vdots & \vdots & \vdots & \\ \alpha_{n} & \cdots & \alpha_{2n-1} & \end{pmatrix}.}

The rank of this matrix is related to Diophantine properties of \alpha.
To see this, we introduce for \beta=\sum_{i=-\infty}^{m}\beta_it^i\in\mathbb{F}_p((\frac{1}{t})) the fractional part \{\beta\}= \sum_{i=-\infty}^{-1}\beta_it^i and the norm \lVert{\beta}\rVert=\lvert\{\beta\}\rvert=q^k, where k\leq -1 is the largest i\leq -1 such that \beta_i\neq 0. Then we have the following result.

If the rank of M_{\alpha,n} is at most r, then there exist a decomposition r=i+j and a monic polynomial P of degree i such that \lVert P\alpha\rVert<q^{-2n+j}. 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 r, then there exist integers i and h such that i+h=r and the i first rows are linearly independent, each of the following n-r rows is a linear combination of them,
and the minor formed of the first i and last h rows and columns is nonzero. Let L_1,\ldots,L_n be the rows of M_{\alpha,n}. Write L_m=(L_m',a_m) where a_m is the last coefficient of L_m and L'_m the row of the first n-1 coefficients. By the above, we have a relation
L_{i+1}=\sum_{m=1}^i c_mL_m. In fact, we shall show that for any j=0,\ldots,n-r-1, we have

L_{i+1+j}=\sum_{m=1}^i c_m L_{m+j}.

This equation holds for j=0. So we argue by induction and assume it holds for any j'\leq j for some j<n-r-1 and prove it for j+1. To start with, the displayed equation implies that
\displaystyle{L'_{i+1+j+1}=\sum_{m=1}^i c_mL'_{m+j+1}.}
Applying the induction hypothesis iteratively, we find coefficients c_m^{(k)} for m=1,\ldots,i and k\leq j+1 such that

\displaystyle{L_{i+1+k}=\sum_{m=1}^i c_m^{(k)}L_{m}}

and
\displaystyle{ L'_{i+1+j+1}=\sum_{m=1}^i c_m^{(j+1)}L'_{m}.}
These coefficients satisfy the initial condition c_m^{(0)}=c_m and the recurrence relations c_m^{(k+1)}=c_{m-1}^{(k)}+c_ic_m^{(k)} for m>1 and c_1^{(k)}=c_h^kc_1.

On the other hand, we know that there exist coefficients d_1,\ldots,d_i
such that
\displaystyle{L_{i+1+j+1}=\sum_{m=1}^i d_mL_{m}.}
Comparing the last equation to the one involving L', and using the linear independence of the first i lines, we find that d_m=c_m^{(j+1)}.

Thus a_{i+1+j+1}=\sum_{m=1}^i c_m^{(j+1)}a_m. But \sum_{m=1}^i c_ma_{m+j+1}=\sum_{m=1}^i c_m^{(j+1)}a_m by definition of the coefficients c_m^{(k)}. We infer that a_{i+1+j+1}=\sum_{m=1}^i c_ma_{m+j+1},
which conclude the inductive argument.

To see the connexion with Diophantine properties of \alpha, notice that L_i is the row of the first n coefficients of \{t^{i-1}\alpha\}. Thus the validity of  the identity for L' for all j=0,\ldots,n-r-1 implies that \{t^i\alpha\}=\{\sum_{m=1}^ic_mt^{m-1}\alpha\}+\beta
where \beta\in\mathbb{T} satisfies \lvert\beta\rvert<q^{-2n+h}.
That is, using the polynomial P=t^i+\sum_{m=1}^ic_mt^{m-1}, we find that \lVert P\alpha\rVert<q^{-2n+h} which concludes the proof of the direct statement. The converse is straightforward.

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