From: "Christopher J. Henrich" Subject: Re: Implicit function with singularity Date: 8 Jan 2000 12:10:11 -0600 Newsgroups: sci.math.research Summary: Weierstrass preparation theorem and rings of power series Xavier Leleu wrote: > I wonder if the following problem is well-known > > Let f be a function R X R --> R which is equal to the sum of its series > f(x,y)= sum over (i,j) non negative integers of a_{i,j} x^i y^j > in a neighborhood of (0,0) and such that f(0,0)=0 > > Is the set of (x,y) in R X R such that f(x,y)=0 equal to a finite union of curves > in a neighborhood of (0,0)? Yes. The general theorems are much easier to handle if you work with complex numbers, but I think the only effect of working with real numbers is that you don't see the curves that go off in non-real directions. The crucial step is the "Weierstrass preparation theorem." It is found near the beginnig of typical texts on "several complex variables." Chris Henrich ============================================================================== From: lrudolph@panix.com (Lee Rudolph) Subject: Re: Implicit function with singularity Date: 8 Jan 2000 12:10:02 -0600 Newsgroups: sci.math.research xavier@math.Hawaii.Edu (Xavier Leleu) writes: >I wonder if the following problem is well-known Yes. "Classical", even. >Let f be a function R X R --> R which is equal to the sum of its series >f(x,y)= sum over (i,j) non negative integers of a_{i,j} x^i y^j >in a neighborhood of (0,0) and such that f(0,0)=0 > >Is the set of (x,y) in R X R such that f(x,y)=0 equal to a finite union of curves >in a neighborhood of (0,0)? Yes (provided you allow the singleton {(0,0)} as "a finite union of curves", if for example f(x,y)=x^2+y^2). It follows, for instance, from material covered in Chapter 5 of R. Narasimhan, _Introduction to the Theory of Analytic Spaces_ (Springer LNM 25, 1966). Lee Rudolph ============================================================================== From: wcw@math.psu.edu (William C Waterhouse) Subject: Re: [Q]: noetherian rings Date: 26 Apr 2000 21:15:53 GMT Newsgroups: sci.math In article <2svh14d592.fsf@ultra2.math.wisc.edu>, Anatoly Karp writes: > Hello all, > > it appears to be folklore that the following rings > are noetherian: > > 1) K[[X]] = the formal power series with coefficients > in the (say) field K > > 2) the subring of K[[X]] consisting of the series > converging in a neighbourhood of zero (or, in other > words, the germs of analytic functions at the origin) > > Can anybody point me to proofs or at least hint as > to what the idea of proofs might be? If this really is the only case needed, and "X" means just one variable, then much more is true: Theorem. For R = either of these rings, the only ideals are the multiples of X^n for various n. Proof. Take any nonzero element of R. We can write it as a power of X times a series with nonzero constant term. But such a series is invertible in R (trivially in the first case, by a standard theorem in the second case). Thus each ideal is generated by the lowest power of X in it. More generally, if A is noetherian, then A[[x_1,...,X_n]] is noetherian; and for K = real or complex field, the convergent power series in n variables are noetherian. The catchword for the proof in the second case is "Weierstrass Preparation Theorem". Both results can be found in Zariski and Samuel, Commutative Algebra, Vol. 2, Chap. VII, Section 1. William C. Waterhouse Penn State ============================================================================== From: Boudewijn Moonen Subject: Re: [Q]: noetherian rings Date: Fri, 28 Apr 2000 17:05:23 +0200 Newsgroups: sci.math Anatoly Karp wrote: > > Hello all, > > it appears to be folklore that the following rings > are noetherian: > > 1) K[[X]] = the formal power series with coefficients > in the (say) field K > > 2) the subring of K[[X]] consisting of the series > converging in a neighbourhood of zero (or, in other > words, the germs of analytic functions at the origin) > > Can anybody point me to proofs or at least hint as > to what the idea of proofs might be? > > Thanks, > -- > Anatoly Karp e-mail: karp@math.wisc.edu > Department of Mathematics URL: http://www.math.wisc.edu/~karp As pointed out by the former repliers the case of one variable is immediate, so the interesting case is of several variables X_1, ...,X_n. This is not quite folklore, I think, at least in the convergent power series case, where it is a theorem in a famous paper of W. Rueckert about 1939, where an extensive study of local analytic algebras was made. Whether the formal power series case is folklore, I do not know; at least it appears in the Zariski and Samuel book on commutative algebra, and a nice proof is due to H. Sagres, J. Reine Angew. Math. 283/284, 436-437. Besides these ad-hoc-proofs one can view these results more conceptually as follows. Let R be any of the following three rings 1) K[X_1, ..., X_s] the polynomial ring 2) K[[X_1, ..., X_s]] the formal power series ring 3) K{X_1, ..., X_s} the convergent power series ring (K suitable) Then R in all three cases is noetherian, and from a modern point of view there is a common cause for this, a combinatorial result known as "Dickson's Lemma". More precisely, the noetherian property is based on the following facts: A) Dickson's Lemma B) A division algorithm based on a chosen monomial ordering (which in case 1) and 2) generalizes Weierstrass' Division Theorem and so avoids clumsy inductive proofs) In more detail: A) Dickson's Lemma ================== In a 1913 paper on perfect numbers in the American Journal of Mathematics (Am. J. Math 35, 1913, 413-423), Dickson proved (Lemma B on p. 415 of loc.cit.) If p_1,...,p_n are given integers, any set S of integers p_1^{e_1}p_2^{e_2}...p_s^{e_s} (e's integers >= 0) contains a finite number of integers F_1,...F_k such that every integer of the set S is a multiple of at least one F_i. A moment's thinking convinces oneself that this is a purely combinatorial property of the exponent vectors E = (e_1, ..., e_s). It can therefore be reformulated as a finiteness property of "semilattices" S in N^s (where N denotes the natural numbers), a semilattice S being a subset closed under addition: E := (e_1, ..., e_n) in S, A := (a_1, ..., a_s) in N ==> E + A := (e_1 + a_1, ... e_s + a_s) in S Then Dickson's Lemma reads For any semilattice S there are finitely many elements E_1,...,E_k in S such that S = \bigcup_{i=1}^k (E_i + N^s) From this it is also clear that there is a canonical set of E_i's, namely those which are minimal w.r.t. the partial order E <= F <==> e_j <= f_j for all j . But points in N^s correspond to monomials in variables X_1,...,X_s, and semilattices correspond to \emph{monomial ideals} in R, this being ideals generated by monomials (since f in R belongs to a monomial ideal iff all its monomials belong to the ideal, a monomial ideal is completely determined by the monomials it contains and hence by the corresponding semilattice of exponents). Thus Dickson's lemma takes the slick form Any monomial ideal in R is finitely generated. A canonical set of generators is provided by those monomials in S which are minimal w.r.t the divisibility relation. REMARK 1) All three interpretations are already in Dickson's paper. 2) In the last form, Dickson's Lemma is a special case of Hilbert's basis theorem for polynomial rings, which is also noted by Dickson. He gives, however, an independent proof, which, as we will see, in turn provides a proof of Hilbert's theorem. A modern reference for the result is the book "Ideals, Varieties, and Algorithms" by Cox, Little and O'Shea. 3) The fact that f in R belongs to a monomial ideal iff all its monomials belong to the ideal is analogous to the similar statement for homogeneous ideal. In fact, the decomposition of f in R into monomials is a refinement of its decomposition into homogeneous components, and ordering the monomials (an idea going back to Macaulay) as we will do below is the basic idea behind the standard basis approach; it is a far generalization of the method of basing algorithms and proofs on the notion of "degree" (a natural number) to basing it on the notion of "monomial degree" (a vector in N^s). B) The division algorithm ========================= Divisibility of monomials is a purely combinatorial notion and can be simply algorithmically performed. In case of one variable, this leads to the well-known division algorithm of dividing one polynomial by another with remainder. This works because each polynomial in one variable has a unique monomial of highest degree, called its \emph{leading monomial}. The algorithm of dividing f by g then proceeds by first dividing the leading monomial of f by the leading monomial of g, multiplying g with the result, subtracting this from f thus obtaining the new f, and repeating this until the leading monomial of f is smaller than the leading monomial of g which then gives the remainder r. Keeping track of the succesive quotient monomials and collecting terms gives the representation f = qg + r . In several variables, there is a priori no leading monomial. To restore this, one introduces an \emph{ordering} on the monomials M(R) := { M := X^A := X_1^{a_1}...X_s^{a_s} | A \in N^ s } i.e. by introducing an order M > M' with (i) > is a total ordering (ii) M > M' ==> MN > M'N for all N in M(R) (iii) > is a well-ordering. There are many such orderings and they have been classified by Robbiano. A feasible representative is the \emph{graded lexicographic order} X^A > X^B :<==> deg X^A > deg X^B or in A - B the leftmost nonzero entry is positive. Given f in R, we define the \emph{leading monomial} LM(f) of f to be the largest monomial in f in case 1), and the smallest monomial in case 2) and 3) for R above. Now, given once and for all, g_1,...,g_k in R, we may divide f by g_1,...,g_k as follows: a) Let i be the first index such that LM(g_i) divides LM(f). Let M be the result b) Replace f with f - Mg_i, call it again f c) Repeat 1) as long as possible. In case 1) this will terminate, since > is a well-ordering. Keeping track of the quotient monomials and collecting terms shows that we can divide a polynomial f by polynomials g_1,...,g_k, obtaining f = f^1g_1 + ... + f^kg_k + r where r is such that no monomial of r is divisible by any LM(g_i). This is elementary. In case 2) and 3) the algorithm will generally not stop; however, the resulting f's will have leading monomials of ever-increasing degree. From this one is able to infer in case 2) that we obtain a representation f = f^1g_1 + ... + f^kg_k + r with f_1,...,f_k in R and r is such that no monomial of r is divisible by any LM(g_i). This is again elementary, given basic properties of the formal power series ring. The delicate question in case 3) is to show that f_1,...,f_k,r, obtained as formal power series, are in fact convergent and so also belong to R. This is true, and the corresponding division theorem is known as "Weierstrass prepare a la Grauert-Hironaka". It is originally due to Grauert (in a paper on deformations of singularities) and Hironaka (somewhere in his reolution of singularities paper, I think). The name is due to Galligo who gave some proofs of it in the 70s (some SLNM), and I myself reported on it in the appendix to the book "Equimultiplicity and Blowing Up" by Herrmann, Ikeda and Orbanz. Anyway, in all three cases of R, we have the Division Algorithm. Now, given any ideal I in R, the basic idea which gets things going is that the \emph{leading ideal} LM(I) := { LM(f) | f in I } of the leading monomials of the members of I contains on one hand a lot of information on I and, on the other hand, due to its combinatorial nature is open to much easier manipulation than I itself. A first manifestation of this principle is the following theorem, which gives us what we want THEOREM Let g_1,...,g_k in I be such that LM(g_1),...,LM(g_k) generate the leading ideal LM(I). Then g_1,...,dots,g_k generate I. PROOF. Let f be in I. By the Division Algorithm, we may write f = f^1g_1 + ... + f^kg_k + r with f_1,...,f_k in R and r is such that no monomial of r is divisible by any LM(g_i). Now r is in I, so LM(r) is in LM(I). Since LM(I) is, by assumption, generated by the LM(g_i), LM(r) is divisible by some LM(g_i) if r is not 0. But no monomial of r is divisible by any LM(g_i). So r = 0. QED. COROLLARY Any ideal I in R is finitely generated, i.e. R is noetherian. PROOF. By Dickson's Lemma, there are g_1,...,g_k such that LM(g_1),...,LM(g_k) generate LM(I). By the theorem, g_1,...,g_k genetate I. QED. So here we are. FINAL REMARK. It is not true in general that, if g_1,...,g_k generate I, the leading monomials LM(g_1),...,LM(g_k) generate LM(I). The reason for this is that there might be a monomial relation N^1LM(g_1) + ... N^kLM(g_k) = 0 between the leading monomials of the g_i (a "monomial syzygy") which does not lift to a relation f^1g_1 + ... + f^kg_k = 0 , LM(f^i) = N^i (a "syzygy") between the g_i. Then the remainder r = f - N^1g_1 - ... - N^kg_k is in I, but may have LM(r) not being divisible by any LM(g_i). So ideal bases g_1,...,g_k having the property that the LM(g_1),...,LM(g_k) generate LM(I) are special and called \emph{standard bases} (or \emph{Groebner bases} in case 1)). The above argument shows that there always exist standard bases of I. This, however, does not settle the question of how to construct algorithmically a standard base of an ideal I, given generators g_1,...,g_k. However, elaborating on the introducing remark on syzygies, one may show THEOREM Let I be an ideal generated by g_1,...,g_k. Then the following properties of g_1,...,g_k are equivalent: (i) g_1,...,g_k is a standard basis of I (ii) Every monomial syzygy of LM(g_1),...,LM(g_k) lifts to a syzygy of g_1,...,g_k. Although it does not appear to be spelled out clearly in the literature, this is the driving force behind the Buchberger algorithm or Mora's tangent cone algorithm for constructing standard bases. Namely, generate in a clever way monomial syzygies of the LM(g_1),...,LM(g_k). By a little miracle, it is the Division Algorithm again which shows you whether they lift to a syzygy of the g_i or not. In case it produces a remainder, join this remainder to the given ideal basis. This terminates in finitely many steps, leaving you a basis with (ii) and hence with (i). Once a standard basis is obtained, all kinds of wonderful things can be done theoretically and computationally. One example of a theoretic application is the above proof of the noetherian property. All in all, the theory of standard bases has revolutionized the subject inthe last 15 years. Regards -- Boudewijn Moonen Institut fuer Photogrammetrie der Universitaet Bonn Nussallee 15 D-53115 Bonn GERMANY e-mail: Boudewijn.Moonen@ipb.uni-bonn.de Tel.: GERMANY +49-228-732910 Fax.: GERMANY +49-228-732712 [slightly reformatted --djr