Some remarks on Fitzpatrick and Flynn's Gröbner basis technique for Padé approximation.
The authors continue work of \textit{P. Fitzpatrick} and \textit{J. Flynn} [J. Symb. Comput. 13, No. 2, 133--138 (1992; Zbl 0746.68045)] exploiting Gröbner bases for obtaining multivariable Pade-approximants. Consider \(h\in k[\underline{x}]=k[x_1,\ldots, x_s]\) representing an initial segment of a Taylor series of a function to be approximated. In the equation lattice approach to multivariate Padé approximation one prescribes finite subsets \(N,D\subset \mathbb Z_{\geq 0}^s\) and a third finite subset \(E\subset \mathbb Z_{\geq 0}^s\) and seeks to determine \(a(x)=\sum_{\alpha\in N} a_\alpha x^\alpha, \) \(b(x)=\sum_{\beta\in N} b_\beta x^\beta\) so that \(a(x)-b(x)h(x)= \sum_{\gamma\in \mathbb Z_{\geq 0}^s \setminus E } u_\gamma x^\gamma.\) To be well posed, one requires certain exponent conditions, like that \(| N| +| D| \geq | E| +1,\) and if \(\gamma \in E, \gamma=\delta+\varepsilon,\) then \(\delta,\epsilon \in E.\) Let \(I=\text{span}_k \{x^\gamma: \gamma\in \mathbb Z_{\geq 0}^s \setminus E \}.\) \(I\) is a monomial ideal in \(k[\underline{x}].\) The elements \((a,b)\) of \(M=\{(a,b)\in k[\underline{x}]^2: a\equiv bh \bmod I \}\) are considered as generalizations of classical Padé approximants. On the module \(k[\underline{x}]^2\) the authors define an order \(> \tau\) as follows: \(x^\alpha e_i >\tau x^\beta e_j\) iff \(| \alpha| >| \beta| \) or \(| \alpha| =| \beta| \) and \(i<j\) or \(| \alpha| =| \beta| \) and \(i=j\) and \(x^\alpha > x^\beta,\) the latter referring to a prefixed monomial order on \(k[\underline{x}]\) [see Ch. 5 of \textit{D. Cox}, \textit{J. Little} and \textit{D. O'Shea} [Using Algebraic Geometry (1998; Zbl 0920.13026) for concepts and notation]. The authors prove that if there exists an up to constant multiples unique \((a_0,b_0)\in M\) such that \(\tau(a_0)< \tau(b_0)\leq m\) then such must also appear in any Groebner basis for \(M\) w.r.t. the \(> \tau\) order. Here \(\tau(.)\) means total degree, and \(m\) is a bound on the degrees of \(D.\) This result is of the same form as Fitzpatrick and Flynn's, but is, due to the choice of their order, wider applicable. A number of examples are given.
scientific article; zbMATH DE number 1967112