A note on the Hopf-Stiefel function

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(1)Article. A note on the Hopf-Stiefel function. ELIAHOU, Shalom, KERVAIRE, Michel. Reference ELIAHOU, Shalom, KERVAIRE, Michel. A note on the Hopf-Stiefel function. Enseignement mathématique, 2003, vol. 49, no. 1/2, p. 117-122. Available at: http://archive-ouverte.unige.ch/unige:12379 Disclaimer: layout of this document may differ from the published version..

(2) L'Enseignement Mathématique. Eliahou, Shalom / Kervaire, Michel. NOTE ON THE HOPF-STIEFEL FUNCTION. L'Enseignement Mathématique, Vol.49 (2003). PDF erstellt am: May 20, 2010. Nutzungsbedingungen Mit dem Zugriff auf den vorliegenden Inhalt gelten die Nutzungsbedingungen als akzeptiert. Die angebotenen Dokumente stehen für nicht-kommerzielle Zwecke in Lehre, Forschung und für die private Nutzung frei zur Verfügung. Einzelne Dateien oder Ausdrucke aus diesem Angebot können zusammen mit diesen Nutzungsbedingungen und unter deren Einhaltung weitergegeben werden. Die Speicherung von Teilen des elektronischen Angebots auf anderen Servern ist nur mit vorheriger schriftlicher Genehmigung des Konsortiums der Schweizer Hochschulbibliotheken möglich. Die Rechte für diese und andere Nutzungsarten der Inhalte liegen beim Herausgeber bzw. beim Verlag.. SEALS Ein Dienst des Konsortiums der Schweizer Hochschulbibliotheken c/o ETH-Bibliothek, Rämistrasse 101, 8092 Zürich, Schweiz [email protected] http://retro.seals.ch.

(3) A NOTE ON THE. by Shalom. HOPF-STIEFEL FUNCTION. Eliahou. and. Michel Kervaire. Introduction volume [P], Alain Plagne gives a formula for the (generalized) Hopf-Stiefel function f3 p Given a prime number p, and two positive integers r, s, recall that (3 p (r, s) In the preceding paper of this. .. is is. defined. as. the smallest integer. the idéal generated by x. r. and. n. /. such that (x + y) n in the. (x. G. ,/),. r. polynomial ring. Fp. where (x r. ,/). [x,y\.. Plagne's theorem reads THEOREM. 1.. Let. r,s. be. positive integers, then. (3. p. (r,s). is. given by the. formula (1). formula is derived Number Theory, Theorem 4, which In. [P],. Hère,. this. we. give. as. is. a. corollary of. a. theorem on Additive. the main resuit of the paper.. another proof of Theorem. 1. using. a. purely arithmetical. argument.. Recall from [EK, p. 22], where f3 p (r,s) was introduced, that this function and s — 1 as can be described in terms of the p-adic expansions of r — 1. follows. *) During the préparation of this paper, the first author has partially benefited from contract with the Fonds National Suisse pour la Recherche Scientifique.. a. research.

(4) THEOREM. r-1. Let. 2.. =. respective p-adic expansions of. /or. Xw>o r —. a. 'V and. 1. s-l. and s. -. 1,. =. with. b. J2t>o <. 0. îP. a^bi. l. the. be. < p. —. 1. a// i.. Define the integer exists. Otherwise, that Then,. fi p. (r,s). k. largest index for which a^ -\+ bip < p — for ail i > 0, set. the. as. if ai determined. is. is. 1. > p,. b^ k. if any. = —1.. by. (2). Although the point of Plagne's paper formula with Additive Number Theory, it admits. a. This. is. to. stress the. is. interesting direct proof using the above Theorem 2.. is. to. relationship of his note that (1) also. the content of the next section. In Section 2, we provide. proof of Theorem. a. simple. 2.. Deriving Theorem. 1.. 1. from Theorem. 2. very easy to understand the relationship of the floor-function [£J or intégral part of £ appearing in Theorem 2, with the ceiling-function [£] the smallest integer at least as big as £, used in formula (1). It. is. ,. ,. ,. The main object of this section will be to locate the minimum over of the expression. attained. at. £. =. (-r +MH—. k+. For every index. Since. r—l+. J2t>o. with. 1. £. as. £. P. show that this. an d to. defined in Theorem. 2.. > 0, we hâve. a iP. Similarly, we hâve 0< (3). k. l). l >. f°U° ws mat. aiP. <. ". =. <1,. and. £. >. minimum. 0. is.

(5) vu. *'\f\. ™c. Applying. for every. =. [ir\. If. a/. minimum of the expression. <p-1. +. yields for. £. at. <. /,. (J j£. |. +. |^| ~^)P. 0. then. (|~£~| + |~^~|. -l)p*isa. weakly. Indeed, the équation. .. < t!. £=0. If there exists A:. />O,. for every. Thus, in the case where. for. +1• Hence,. £.. increasing function of £>. attained. \yr\. =. Je. £. function of. a. -. the same formulas to s, we hâve. It remains to locate the as. l. +. k= -1. and min£> an. index. the. {(h^. 0. k. ,. minimum of (\ +. I. > 0 such that ak-\-b. )^} k^p. then the above calculation shows that. weakly increasing function of On the other hand, for. £. £. for. (. -Ijp£is. £+ =. r. and -^. +' y " I>as desired. 0. < a;+fc/ < /?—. +. —. l)p. £. is. 1. a. +I<£.. < k, we hâve. Therefore, even though the function. monotonously decreasing in the interval gênerai, it still does take its minimum at. (. + MH —11. ~r. <. 0 £. =. £. < k,. k+l.. //. need not be. and it actually is not in.

(6) Consequently, in both cases. Now, Theorem. and Theorem. 2. 1. tells. noted. in. Similarly,. -1. =. and k > 0, we hâve. that. us. follows.. Proof of Theorem. 2.. As. k. équation =. E;>*. V~. +1. Section. of. (3). (. 2. =. hër. 1,. s];>£+i. <^~(~. (^. +1) .. +1). *. .. By définition of k, we hâve right hand side of the équation. ai. +. bt. < p. p-adic expansion of the left hand side. For the purpose of the proof of Theorem. —. for. 1. z. >. k+. 1. and thus the. the. is. 2,. set. (4). We (x + v). n. proceed to show that w+ p k+l is the smallest integer n such that s r r belongs to the idéal (% ,/) = x ¥ p [x,y]+y F p [x,y] in the polynomial. ring ¥ p [x, y] (x. r. .. That. is. w+. We. first calculate. ,/).. We hâve from (4). k pp. +. (x + y). l. w. =. op(r,0. in. p. (r, s). .. quotient algebra of ¥ p [x,y]. the. We claim that (5). modulo. (x r. ,/),. where. u= Ei>*+i. a iP. l. and. v=. J2t>k+i. b tP. l. -. modulo.

(7) since. Indeed,. expansions of. ,. t. +bt. c= Y,i>k+i. expressions. ai. a. < p l. c iP. -. and. for. 1. i. >. k+. d= Yli>k+\( a. i. 1. +. définition of. by ~. C. is. not equal to. —. g. >. Z?^. and. i>£+l.. Thus in this case the monomial x c y d belongs to the idéal. If, on the contrary,. (x + y). w. is. and. ci >ag. =a^ for. c/. r. s. (x ,y ).. />£+l,. this implies. indeed given by formula (5) modulo (x r ,y s ).. Now, observe that the product of binomial coefficients 7 = IT^jt+i C'^O non-zero in ¥ p and we can write (x + y) w =7• x u y v modulo (x r ,y s ). It is now easy to finish up the proof of the theorem. //+. However,. +u=l+. 1. +1. +v>s. Summarizing, (x +yf. Similarly,. ,/. using (x +. +l. -. 1. £?=o(P - 1)^ + J2i> k +i +w. G. =. {x. =. r. \y. s. ). "&. :. >I+(r-I)=r.. and thus. EÛ'-^-iyy^'-^. 1. in. F. p. [x,,].. It is immédiate to see that, calculating modulo (x and with the notation = Z)/=o a i and = Z)i=o h we can restrict the summation over the r. uo. the P~ âdic. are. respectively. If for a given c, there is an index i>k-\-l for which g dénote by £ the largest / such that g a^. If c^ < ag and g = a/ for i > £+ l, this implies <^ + bi ~~ c Therefore we hâve + —bi for. Thus. the. and d. c. i. is. *V. k,. interval. ,. k. pp. +. l. -1-vo<j<uo:. ,/),. jto.

(8) Moreover, the monomials appearing on the right hand side are distinct, hâve non-zero coefficient ±7 and form a non-empty subset of an F^-basis r of view of ,/). Indeed, on the one hand, p k+l p [x,;y]/(x. -l-vq<uoin. Fp[x,;y]/(F. the inequalities. j+u <u +u. and on the other hand. If. = —1, then. k. u§. Summarizing. o. =. vq. =. 0. =. r-l. and. pk+l-j-1p. k+l. -j-. and the above conclusion. 1. +v <. vo. +v =. s—. 1 .. still holds.. :. and this complètes the proof of Theorem. 2.. REFERENCES [EK]. ELIAHOU, /.. [P]. PLAGNE,. S.. and M.. Kervaire. Sumsets. in vector spaces. ofNumber Theory 71 (1998), 12-39. A. Additive number theory sheds new light. over finite fields.. on the. Hopf-Stiefel. o. fonction. L'Enseignement Math. (2) 49 (2003), 109-116. (Reçu le 31 janvier 2003). Shalom Eliahou. Département de Mathématiques LMPA Joseph Liouville Université du Littoral Côte d'Opale Bâtiment Poincaré 50, rue Ferdinand Buisson, B.P. 699 F-62228 Calais France. e-mail. :. eliahou. @. lmpa.univ-littoral.fr. Michel Kervaire Département de Mathématiques Université de Genève 2-4, rue du Lièvre B.P. 240. CH-1211 Genève 24 Suisse. e-mail. :. [email protected].

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