A GENERALIZATION OF GAUSS THEOREM ON QUADRATIC FORMS

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1 A GENERALIZATION OF GAU THEOREM ON QUADRATIC FORM Nicole I Brtu d Adi N Cret Deprtmet of Mth - Criov Uiversity, Romi ABTRACT A origil result cocerig the extesio of Guss s theorem from the theory of biry qudrtic forms over forms with more ukows ws preseted by Brtu i 994 The result did ot pper i world-wide publictios d it ws ot sufficietly exemplified i pplictios This is the purpose of the ctul pper Keywords: Represettio of elemets; qudrtic forms; Guss s theorem; utomorphic trsformtio; the Pell s equtio, Lgrge s Four-qure Theorem Actul theory A qudrtic form over rbitrry field K of chrcteristic ot is homogeeous polyomil hvig the coefficiets i K : f = ijx jxi(ij ji) or f = X' A X () The represettio of elemets of K implies the solvbility of the equtio f =0 d the fidig of lgorithm for the represettio of zero Theorem Mikowski-Hsse resposes to the first coditio [] The theory solves the cse of qudrtic forms of rk for the secod coditio There is isomorphism betwee the set of clsses of similr modules d the set of clsses of biry qudrtic primitive d self equivlet forms The fudmetl otio is the utomorphic trsformtio, defied s the lier trsformtio of determit D =, which trsforms qudrtic form ito itself I the cse of rtiol fields, Guss stted remrkble result, which defies the coefficiets of the uimodulr mtrix of the trsformtio

2 Guss s Theorem For biry qudrtic form: f (x, y) = x + bxy + cy, where we suppose (,b,c) =, if the lier trsform of mtrix G = is utomorphic, the: t bu ; cu () t bu u ; where t, u Z, such tht they verify the Pell s diophtic equtio: t - du = 4 (3) d d is the discrimit of the form The coverse of this theorem holds too The proof of it showed i [] Guss s theorem reduces the quest for solutio of the diophtic equtio f(x,y) = m, to the fidig of solutio of the Pell s equtio (3) For the pplictios, [], we hve defied the miiml iteger positive solutio differet from the ordiry solutio (t =, u =0), s beig the ordered pir (x, y), with x>0 d y>0, for which, for y other solutio (x`, y`), we hve x<x`, or, if x=x`, the y<y` A geerl theorem of the Guss type I [4], we showed tht we could ffirmtively respod to Dickso s problem [3] for the secod-degree equtios with severl ukows proposig geertig method for rtiol solutios The completio cocerig the umeric represettio is give i [5] Oe of the pplictios of the geerlized theorem ws discussed i [6] The result ppered i [4] is developed d exemplified i the preset pper Let f be qudrtic form of severl ukows, which is esier to be writte i the coicl form: f = x + + i x i - i+ x - - i x [4] where, N The determit will be: d = (-) -i d the lgebric sum of the coefficiets will be clled the trce of the digol qudrtic form: = + + i - -

3 Remrk = 0, if = 0, vrible chge is mde, so tht we hve =0 Geerl Theorem (Brtu) For y rtiol qudrtic form [4], brought to the coicl form, oe c determie utomorphic lier trsformtio defied by the uimodulr mtrix below: B = i i i i where,, N, the coefficiets of the form, d Z is the trce of the qudrtic form For the biry qudrtic form, the mtrix B is ideticl to Guss s mtrix G, built for the rtiol solutio of Pell s equtio ttched to the form If the equtio f = m, with d positive d m rtiol, hs solutio, the the mtrix B geertes the set of rtiol solutios Proof The mtrix B comes from the mtricil writig of the geertig reltios of positive icresig solutio for the x vrible, from turl solutio (x, x ): x ' = - x + ( x + i x i - i+ x i+ - - x ) (6) x ' = -x + ( x + i x i - i+ x i+ - - x ) Chgig the sig i the colums i +,, of the mtrix we write: B = I - A (7) with det B = (-) -i det B d det B = det B (8) 3

4 where I is the uit mtrix of -order, det B is the determit of the mtrix B d A is degeerte mtrix of the form: A I order to show tht B is uimodulr, we tke: B = ( I 4 I A + 4 A ) d we get B = I, i e det B = For the secod prt of the theorem let x - by = m (9) be qudrtic equtio, with b > 0 We hve: d = 4 b d = b (0) The mtrix B is writte : B = - b b () d B = b b b b (`) with det B = A rtiol solutio of Pell s equtio is: b t o = b d u = b 4

5 Guss s mtrix built from this solutio: G = b b b b () We get: G B qed (`) Remrk I pplictios, for the geertig of other turl solutio {x } from oe tht is give {x}, it is ecessry d sufficiet tht the term: t = - ( x + - x ) (3) hs t lest iteger vlue mog the - possible vlues A sufficiet coditio is tht, the trce of the qudrtic form, to hve the vlues or If ot, we use power mtrix B, which is still uimodulr It is ecessry extesio of the previous defiitio of the miiml solutio Defiitio For the equtio ttched to the -degree qudrtic form, we cll miiml positive solutio the ordered (x, x ) which is solutio of the equtio i which ll vribles re itegers d oegtive d t lest oe is ot ull d verifies: for y other ordered {x ' }, rtiol iteger oegtive solutio, i we hve x < x ', if x = x ', the x < x ', d if x -= x ', the x - < x ' Cosequece The multitude of the iteger solutios of the equtio () c be represeted by the odes of lttice The utomorphic trsformtio of mtrix B geertes the multitude of solutios: X i+ = X i *B (4) The sigs for the itegers x i re choose so tht iteger solutios c be obtied x i+ 5

6 3 Exmples 3 The equtio x - y = 7 I the literture, the equtio is exmple for the solvig method of biry equtios with positive determit, by the ctul method; we hve Pell s equtio: t - 8u 3 4 = 4 d Guss s mtrix G = 3 We cosider the couples (3, ) d (5, 3) s the miiml positive solutios d by recurret formuls: x i+ = 3x i + 4y i y i+ = x i + 3y i we get two ifiities of turl solutios By the proposed method, we pply the trsformtio defied by B mtrix, where = - : 3 4 B = = G 3 Accordig to defiitio, the miiml positive solutio is uique, mely (3,) The grphic represettio of the solutio set is chi, the top beig the miiml solutio From the miiml solutio the chi top- (3,) there c be obtied two mjor solutios: (5,3) d (3,9), etc 3 The equtio x - 5y = I literture, this equtio of type Pell is solved through the fidig of the miiml positive solutio, usig the method of cotiues frctios [] I our method, Pell s equtio does ot hve specil role ymore trtig from the ordiry solutio (,0), the use of the mtrix B, where = -4, will geerte frctiol solutios We look for turl umber p, so tht the mtrix B could hve oly iteger elemets We hve: B = B = d the iteger solutios will be geerted by this 4 9 mtrix B : (x, y ) = (9,4), etc 3 3 The equtio x + y + 3z - 5w = 5 A ordiry equtio where = 6

7 Mtrix B is writte: B= trtig from certi solutio = (3,,, ), other 5 ew solutios c be obtied: (5, 4, 3), etc 3 4 The equtio x + y + z = 89 The equtio hs solutios, the umber 89 beig ot equl to 4 l (8k + 7) Let = (9,,) be the give solutio d we check if there is t lest rtiol iteger vlue t mog the possible four: We build the mtrix B: t = 3 ( 9 ) B= 3 d we determie the solutios : (3,48), (0,58), (6,,7), which re ll the decompositio i sum of three squres We otice tht 89 is prime umber hvig uique decompositio i sum of two squres 3 5 The equtio x + y + z = w This equtio ws discussed i [6] The mtrix B is writte: i+ = i * B d B = The multitude of solutios is represeted through the odes of grph, with the top the ordiry solutio (,0,0,) 7

8 I [7], we showed tht, usig the fuctio qudrtic combitio, we fid the geerl solutio of the diophtie equtio, type Euler- Crmichel-Mordell: x + b y + c z = w (b, c) Z We eucited theorem, which is stroger th the Lgrge s Four qure Theorem: Theorem (Brtu) For y turl umber z, there re t lest three iteger umbers (u,v,w), or/d (,b,c), i order to hve represettios: z= u + v + w ( ) z= ² + b² + c² ( ) (5) For z = z = k (8l + 7) we hve oly the represettio ( ), for z = z = k+ (8l + 7) we hve oly the represettio ( ) d for z z z we hve, i the sme time, the represettios ( ) d ( ) Exmples: z = 5 we hve z = * ( ) z = 30 we hve z = ( ) z = we hve z = ( ) d z = * ( ) Coclusio The proposed method, for the determitio of the solutios of qudrtic equtios, is differet from the oes tht exist i literture, from Fermt, Lgrge, Guss # # # REFERENCE HHAE Uber die Drstelbrkeit vo Zhle durch qudrtische Forme i Korper der rtiole Zhle J reie gew Mth 5 (9) p9-48 LJMORDELL Diophtie Equtios (909)-Acdemic Press Lodo Ad New York, cp7-8, p LEDICKON History of theory of umbers t3, p37-39, Chelse Publ Comp(954) 4 NIBRATU Eseu supr ecutiilor dioftice-- Editur Adel Criov (994) p3-7 5 NIBRATU Note de liz dioftic- Editur Dutescu Criov (996) p-6 6 NIBRATU Diophtie equtios The first itert cof i umbers theory Americ Res Press (997) p NIBRATU d BNBRATU O the quterry qudrtic diophtie equtios () (000) Bulleti of Pure d Applied cieces Vol 9E/, p

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