Polynomials over Power Series and their Applications to Limit Computations (lecture version) Marc Moreno Maza University of Western Ontario IBM Center for Advanced Studie September 17, 2018 Plan 1 From Formal to Convergent Power Series 2 Polynomials over Power Series Weierstrass Preparation Theorem Properties of Power Series Rings Puiseux Theorem and Consequences Algebraic Version of Puiseux Theorem Geometric Version of Puiseux Theorem The Ring of Puiseux Series The Hensel-Sasaki Construction: Bivariate Case Limit Points: Review and Complement 3 Limits of Multivariate Real Analytic Functions At isolated poles for bivariate functions Limit along a semi-algebraic set At isolated poles for multivariate functions Proof of the main lemma 4 Computations of tangent cones and intersection multiplicities Tangent Cones Iintersection Multiplicities Plan 1 From Formal to Convergent Power Series 2 Polynomials over Power Series Weierstrass Preparation Theorem Properties of Power Series Rings Puiseux Theorem and Consequences Algebraic Version of Puiseux Theorem Geometric Version of Puiseux Theorem The Ring of Puiseux Series The Hensel-Sasaki Construction: Bivariate Case Limit Points: Review and Complement 3 Limits of Multivariate Real Analytic Functions At isolated poles for bivariate functions Limit along a semi-algebraic set At isolated poles for multivariate functions Proof of the main lemma 4 Computations of tangent cones and intersection multiplicities Tangent Cones Iintersection Multiplicities Formal power series (1/4) Notations K is a complete field, that is, every Cauchy sequence in K converges. K[[X1;:::;Xn]] denotes the set of formal power series in X1;:::;Xn with coefficients in K. e These are expressions of the form Σe aeX where e is a multi-index e e1 en with n coordinates (e1; : : : ; en), X stands for X1 ··· Xn , jej = e1 + ··· + en and ae 2 K holds. e For f = Σe aeX and d 2 N, we define P e (d) P f(d) = jej=d aeX and f = k≤d f(k); which are the homogeneous part and polynomial part of f in degree d. Addition and multiplication For f; g 2 K[[X1;:::;Xn]], we define f + g = P (f + g ) and fg = P Σ (f g ) : d2N (d) (d) d2N k+`=d (k) (`) Formal power series (2/4) Order of a formal power series For f 2 K[[X1;:::;Xn]], we define its order as minfd j f 6= 0g if f 6= 0; ord(f) = (d) 1 if f = 0: Remarks For f; g 2 K[[X1;:::;Xn]], we have ord(f + g) ≥ minford(f); ord(g)g and ord(fg) = ord(f) + ord(g): Consequences K[[X1;:::;Xn]] is an integral domain. M = ff 2 K[[X1;:::;Xn]] j ord(f) ≥ 1g is the only maximal ideal of K[[X1;:::;Xn]]. k We have M = ff 2 K[[X1;:::;Xn]] j ord(f) ≥ kg for all k 2 N. Formal power series (3/4) Krull Topology Recall M = ff 2 K[[X1;:::;Xn]] j ord(f) ≥ 1g. Let (fn)n2N be a sequence of elements of K[[X]] and let f 2 K[[X]]. We say that (fn)n2N converges to f if for all k 2 N there exists N 2 N s.t. for all k n 2 N we have n ≥ N ) f − fn 2 M , (fn)n2N is a Cauchy sequence if for all k 2 N there exists N 2 N s.t. k for all n; m 2 N we have n; m ≥ N ) fm − fn 2 M . Proposition 1 We have T Mk = h0i, k2N If every Cauchy sequence in K converges, then every Cauchy sequence of K[[X]] converges too. Formal power series (4/4) Proposition 2 For all f 2 K[[X1;:::;Xn]], the following properties are equivalent: (i) f is a unit, (ii) ord(f) = 0, (iii) f 62 M. Sketch of proof This follows from the classical observation that for g 2 K[[X1;:::;Xn]], with ord(g) > 0, the following holds in K[[X1;:::;Xn]] (1 − g)(1 + g + g2 + ··· ) = 1 Since (1 + g + g2 + ··· ) is in fact a sequence of elements in K[[X1;:::;Xn]], proving the above relation formally requires the use of Krull Topology. Abel's Lemma (1/2) Geometric series From now on, the field K is equipped with an absolute value. The geometric e series Σe X is absolutely convergent provided that jx1j < 1;:::; jxnj < 1 all hold. Then we have Σ xe1 ··· xen = 1 : e 1 n (1−x1)···(1−xn) Abel's Lemma e n Let f = Σe aeX 2 K[[X]], let x = (x1; : : : ; xn) 2 K , let M 2 R>0 and Let ρ1; : : : ; ρn be real numbers such that e n (i) jaex j ≤ M holds for all e 2 N , (ii)0 < ρj < jxjj holds for all j = 1 ··· n. Then f is uniformly and absolutely convergent in the polydisk n D = fz 2 K j jzjj < ρjg: In particular, the limit of the sum is independent of the summand order. Abel's Lemma (2/2) Corollary 1 e Let f = Σe aeX 2 K[[X]]. Then, the following properties are equivalent: n (i) There exists x = (x1; : : : ; xn) 2 K , with xj 6= 0 for all j = 1 ··· n, s.t. e Σe aex converges. n e (ii) There exists ρ = (ρ1; : : : ; ρn) 2 R>0 s.t. Σe aeρ converges. n e (iii) There exists σ = (σ1; : : : ; σn) 2 R>0 s.t. Σe jaejσ converges. Definition A power series f 2 K[[X]] is said convergent if it satisfies one of the conditions of the above corollary. The set of the convergent power series of K[[X]] is denoted by KhXi. Remark It can be shown that, within its domain of convergence, a formal power series is a multivariate holomorphic function. Conversely, any multivariate holomorphic function can be expressed locally as the sum of a power series. ρ-norm of a power series Notation n e Let ρ = (ρ1; : : : ; ρn) 2 R>0 . For all f = Σe aeX 2 K[[X]], we define e k f kρ = Σe jaejρ : Proposition 3 For all f; g 2 K[[X]] and all λ 2 K, we have k f kρ = 0 () f = 0, k λf kρ = jλj k f kρ, k f + g kρ ≤ k f kρ + k g kρ, If f = Σk≤d f(d) is the decomposition of f into homogeneous parts, then k f k = Σ k f k holds. ρ k≤d (d) ρ If f; g are polynomials, then k fg kρ ≤ k f kρk f kρ, limρ!0k f kρ = jf(0)j. Convergent power series form a ring (1/5) Notation n Let ρ = (ρ1; : : : ; ρn) 2 R>0 . We define Bρ = ff 2 K[[X]] j k f kρ < 1g Theorem 0 The set Bρ is a Banach algebra. Moreover, if ρ ≤ ρ holds then we have Bρ0 ⊆ Bρ. S We define KhXi := ρBρ KhXi is a subring of K[[X]]. Cauchy's estimate e e Observe that for all f = Σe aeX 2 K[[X]], we have for all e 2 N kfkρ jaej ≤ ρe : Convergent power series form a ring (2/5) Theorem 1 The set Bρ is a Banach algebra. Moreover, 0 1 if ρ ≤ ρ holds then we have Bρ0 ⊆ Bρ, 2 S we have ρBρ = KhXi. Proof (1/3) From Proposition 3, we know that Bρ is a normed vector space. Proving that k fg kρ ≤ k f kρ k g kρ holds for all f; g 2 K[[X]] is routine. Thus, Bρ is a normed algebra. To prove (1), it remains to show that Bρ is complete. (j) e Let (fj)j2N be a Cauchy sequence in Bρ. We write fj = Σe ae X . n From Cauchy's estimate, for each e 2 N , for all i; j 2 N we have (j) (i) k fj − fi kρ jae − ae j ≤ ρe . Convergent power series form a ring (3/5) Proof (2/3) n (j) Since K is complete, for each e 2 N , the sequence (ae )j2N converges to an element ae 2 K. e We define f = Σe aeX . It must be shown that (i) f 2 Bρ holds and (ii) limj!1 fj = f holds in the metric topology induced by the ρ-norm of the normed vector space Bρ. Hence we must show that (i) k f kρ < 1 holds, and (ii) for all " > 0 there exists j0 2 N s.t. for all j 2 N we have j ≥ j0 ) k f − fj kρ ≤ ". Let " > 0. Since (fj)j2N is a Cauchy sequence in Bρ, there exists j0 2 N s.t. for all j ≥ j0 and all i ≥ 0 we have P (j+i) (j) e " e jae − ae jρ = k fj+i − fj kρ < 2 : Convergent power series form a ring (4/5) Proof (3/3) n (i) Let s 2 N be fixed. Since for each e 2 N the sequence (ae − ae )i2N converges to 0 in K, there exists i0 2 N s.t. for all j ≥ j0 and all i ≥ i0 we have Ps (j+i) e " jej=0 jae − ae jρ < 2 : Therefore, for all j ≥ j0 and all i ≥ i0 we have Ps (j) e jej=0 jae − ae jρ ≤ Ps (j+i) e P (j+i) (i) e " " jej=0 jae − ae jρ + e jae − ae jρ < 2 + 2 = ": Since the above holds for all s, we deduce that for all j ≥ j0 P (j) e k f − fj kρ = e jae − ae jρ ≤ "; which proves (ii). Finally, (i) follows from k f kρ ≤ k f − fj0 kρ + k fj0 kρ ≤ " + k fj0 kρ < 1: Convergent power series form a ring (5/5) Corollary 2 KhXi is a subring of K[[X]].