Analytic Function Theory, Volume I - 2nd Edition (AMS by Einar Hille

By Einar Hille

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A, are linearly dependent over the field of fractions F of A. Writing the above polynomials as vectors in fn+m^ the determinant of the matrix whose rows are these vectors is precisely R Y ( f , g } . Therefore the result follows immediately. 18 Let P,Q e K[Y] be pseudo-polynomials with respect to the indeterminate Y. The series P and Q admit a non-unitary common factor in ^[[Y]} if and only ifRY(P,Q) = 0. 13), P and Q admit a non-unitary common factor in 7£[[F]], if and only if they admit a non-unitary common factor in 7£[F].

24 Hefez From the above Corollary we obtain the following Corollary of the Weierstrass Preparation Theorem. 8 Let FeU\{Q} be of multiplicity n. There exist a K- automorphism T of U, a unit U e U and A} , . . ; = ! , . . , n, and T ( F ) . [/ = A? ~l + ••• + Ar, . The "polynomial" X? ~2 H ---- 4- An associated to F, after possibly a linear- change of coordinates, will be considered a preparation of F for its study. The study of a polynomial is much simpler than the study of a power series. More generally, given a finite number of non-invert ible series in 7£, we have seen that there exists a change of coordinates that allows to prepare simultaneously all these series.

We will denote in what follows by 72/ the ring K[[X\^ . . , -AT r _i]] and by A4n' its maximal ideal. 3 (The Division Theorem) Let F e Mn C K, regular of order m with respect, to Xr. , there exist Q G 72. and R E 7i'\Xr} with R = 0 or deg Yr (-R) < m, uniquely determined by F and G, such that R. , Xr], not necessarily homogeneous, with Ptj = 0 or degXr Rj < rn, for all i, such that mult(gj) > z and mult(P^) > 1 + i, in such a way that if we put Q = qo + qi H , and R = R_i + RQ + R1 H , we will have G = FQ + R.

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