A Gröbner basis technique for Padé approximation
Let \(a,b,h\in R=k[x_ 1,\dots,x_ n]\) and let \(I=(p_ 1,\dots,p_ r)\) be an ideal in \(R\) where the given basis is a Gröbner basis. The solutions \((a,b)\) of the congruence \(a\equiv bh \mod I\) are considered as generalizations of classical Padé approximants. One basis for the solution module \(M\) is \(\{(h,1),(Lt(p_ j):\;1\leq j\leq n\}\) where \(Lt(p)\) denotes the leading term of \(p\). Suppose a solution \((a,b)\) exists with \(a\) and \(b\) relatively prime and in normal form modulo \(I\), and satisfying a weak term order condition: \(Lt(a)\leq\varphi\), \(Lt(b)\leq\psi\) and for all \(\rho\), \(\sigma\) with \(\rho\leq_ T\varphi\), \(\sigma\leq_ T\psi\) and \(\rho,\sigma\not\in(Lt(I))\) the product \(\rho\sigma\) does not lie in \((Lt(I))\). (Here \((Lt(I))\) is the ideal generated by the leading terms of the elements of \(I\), \(\varphi\), \(\psi\), \(\rho\), \(\sigma\) are terms and \(<_ T\) is a term order.) Then \((a,b)\) is a minimal (reduced) solution relative to the term order \(<_ w\) on \(R^ 2\) induced from \(<_ T\) and the weight vector \(w=(\psi,\varphi)\). A scalar multiple of \((a,b)\) appears in any Gröbner basis of \(M\) relative to \(<_ w\). Such a Gröbner basis can be derived from the one given above by changing the term order.
scientific article; zbMATH DE number 56191