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GERARDAS ŽILINSKAS 21 2. 29606404 Lietuvos Valstybinės Leidyklos 1941 m. leidinys Nr. 27- Atsak. redaktorius V. Peironis. Tiražas — 3.200 egz. Spaudė Valstylinė „Varpo“ sp. Kaune, Gedimino 38. M. K. TpeGenwa m C. E. Jiaumi APHOMETMKA IMocoGue AA4 …
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22 GERARDAS ŽILINSKAS Nuorodos 1. C. A. Rogers. Harald Davenport. In: The collected works of Harold Davenport, ed. by B. J. Birch, H. Halberstam, C. A. Rogers. Vol I. Academic Press, 1977, XII-X. . G. Žilinskas. On the class number of indefinite guadratic …
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GERARDAS ŽILINSKAS 23 ON THE CLASS NUMBER OF INDEFINITE GUADRATIC FORMS IN 2 VARIABLEĖS WITH DETERMINANT +1 G. ZILINSKAS*. [Exiracted from the Journal of the Eond:n Mathematical Society, Vol. 13, 1938] 1. In this paper, we deal with guadratic forms AS (1) …
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24 GERARDAS ŽILINSKAS similar result for ten variables: Aj,,=2. Ko* has proved that 4, ,=2 and, that Ian = kiai z3. In the same paper, he proved that there exist improperly primitive, or evenį, definite forms in 2 variables with A= 1-1 if and only if n=0 …
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THE CLASS NUMBER OF INDEFINITE GUADRATIC FORMS 25 I prove further that if A= 1 and 2 = 4, S, and also if A= —1 and m= 6, tliere exists in each case only one even class of forms and this is represented by (6) Ee o 5 )— UAB P Further, for A= 1 and m = 12, …
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26 GERARDAS ŽILINSKAS We derive the positive form E 3 š (10) Jia, Tr, 15)= UL = L 2 bi Zr2. i Evidently this form has determinant d = +-A= I. Call f(z,, 74, 74) reduced if, and only if, f(z,,-1,, 15) is reduced. Hence, making a unimodular transformation …
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THE CLASS NUMBER OF INDEFINITE GUADRATIC FORMS 27 forms Fua, 25) =257-257, Filzo, 15)=157—25, El(tp, 15) — 222 Ta, and hence k(y;, 2; y5) and also f(2,, 23, 75) is transformable into one of (15a) J (tu Tr 15) = L-247-257-733, (156) Ji“ (tas Tą, T3) — …
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GERARDAS ŽILINSKAS we see that g(y;, Y-, y5) can be transformed either into Ra (Zi 22 25) — 22, 22-207 257, or into Ni5 (21, 22, 25) = 22, 25235, according as b,„ is odd. or even. Now we see that A, (24, 22, 24), by the transformations 0 0 1 0 1 ON > ( 0 …
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THE CLASS NUMBER OF INDEFINITE GUADRATIC FORMS 29 where the v's are defined in (4). The determinant of the form is A=|a;|= *1. We derive the positive definite form Ė 4 Jūs Ta, Ta, TV) = VŽ -05 = E TiTj ųI= with determinant d. Obviously d= +1A—=1. If f(zi, …
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30 GERARDAS ŽILINSKAS be either an indefinite or a definite form. If it is an indefinite form, then by Lemma 1, it can be transformed into one of (222) Ailys Vs V) =VU4V5—V44 FalYas Yas Ya) = Y57—V57—V47- If 6(y-, 5, y4) is a definite form, then, by the …
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THE CLASS NUMBER OE INDEFINITE GOUADRATIC FORMS 31 It is easy to see that both forms k(z4, Z2; Z3, 24) are eguivalent to I (Z1s Za, 235 24) = 2 2—25- 623, 24). Also, by replacing 24, z, by —2,, —2,, If necessary, we can omit the minus sign in A (z4, 22, …
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32 GERARDAS ŽILINSKAS The forms are eguivalent to one and only one of the n—1 forms T n * (26) a 2 LS P (GC 2 1 n-:71+1 Proof. The lemma is true for 2, 3, and 4 variables, so let us suppose that it is true for 2—1 variables. Then let n (27) "Ac "gu (a; = …
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THE CLASS NUMBER OF INDEFINITE GUADRATIC FORMS 33 The form (28) can be written as (284) g.(Vis + Yi)= Va(-E2y A BanVa-2baaYs TA 202 Yn) T. Ž bsVV; = Ya(4-2y14-0229> --2032914- 2642 Y41----2-20x2Ya) TĖ1 (Ya Vn): We transform (28a), taking (31) Vi= A usa …
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GERARDAS ZŽILINSKAS is a properly primitive indefinite form in 2— 1 variables with determinant + landindex 7. By hypothesis, it is eguivalent to nT n (36) 2 S Zia 2 n—-41 Hence by (35) and (32) i 5 Z.)— F.(2ų, "445 Z,). (3) 6-1(23: ---, Z,) is positive …
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THE CLASS NUMBER OF INDEFINITE OUADRATIC FORMS 35 7=1 (mod 2), 7= 1,8, ...„2—l. Hence Ci „= 42. (C) "n=1 (mod 2 A= I 7=0 (mod 2), 7=2, 4, ...„n—1. Hence C, ,= į(2—1). (D) n=1 (mod 2), A=—1. 7=1 (mod 2), += 1,3, ...,m—2. Hence C, „= į(2—1). Hence Lemma 3 …
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36 (GERARDAS ŽILINSKAS form in an even number of variables. Hence there exist even forms in mn variables when 2=0 (mod 4), A= I, and also when n=2 (mod 4), A=-l. One part of the statement is proved. We now prove that even forms do not exist if (a) n=0 …
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GERARDAS ŽILINSKAS forms with determinant —1 and an even number of variables do not exist. Hence į(25, ..., 7,) is indefinite. We can now apply the same process to į(z5, ..., 1,) and finally come to the case n = 4, where we know that there is no even form …
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38 GERARDAS ŽILINSKAS For 2 = 8, by the result in six variables with A — —1, we immediately deduce from (41) that there exists only a single class of even forms, repre- sented by (6). The class number C5 „= ž(2—1)-1 = 4. Similarly, for 2 = 12 or 16, we …
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GERARDAS ŽILINSKAS 39 On the product of four homogeneous linear forms. By G. Žilinskas. (Manchester). 1. Let L,,L,,£,, L, be four real homogeneous linear forms in v,, U,, V3, V, with determinant 1. Let (1) M=min |L,L,L,L,, where min denotes the lower …
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(GERARDAS ZŽILINSKAS 1 5 due to Huzwitz, and many different proofs of this have been given, 2. By detinition, there exist for arbitraily small => 0 integral values of the vs s:ch that the corresponding values L,*,... of L,,... The best possible result for …
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ON THE PRODUCT OF FOUR HEMOGENEOUS LINEAR FORMES 41 Thirdly 0, VV, V, VV are not coplanar, and finally 0, V? 70, 8 ya are not in a hyper - plane. Let g) be the values of š, corresponding to V? and put 4 (5) 45= X [EP]. By Minkowski's second main theorem, …
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42 (GERARDAS ŽILINSKAS (IV) Three are negative and one positive, (V) Two are negative and two positive. We prove that in the first 4 cases, (10) Uxz—1)(y—1) (2—1) (w41)|> 1—e. In case (I), by (85) |(a—1)(y—1)(z—1)(w—1)|-> > 1—5, and a Joriiori (10). In …
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ON THE PRODUCT OF FOUR HEMOGENEOUS LINEAR FORMES 43 (14) K=minlx4y4z 0) =27/5 (=4-47...). Proof. We show lHirst that min f(a, y,z,10) > 2 5. Multiplying (b') and (b“) together, we get (b) | (22 —1) (yž— 1) (22— 1) (02 — 1) =1, and we investigate the …
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44 GERARDAS ŽILINSKAS On using Lagrange's method, as in case 1, the minimum in the second. case is attained when y=z=w and when the eguality sign holds in (b). Hence —— F.4)=x413V1 V —=- Hence K,= min /(x,y,z,w) = mini min f(x, y, z, w) (a), …
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ON THE PRODUCT OF FOUR HEMOGENEOUS LINEAR FORMES 45 Then = Žikis IB Ass LL 0 UE Putting again x +-y -z=3/, we have f(x, y, Zz, w)=31/41w. By Lagran- ge's method, it is easily proved that (EU E EL: Hence = F.(f)= min f(x,y,z,w) =31 i ——-3 4 niajjl y, Z, W) …
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46 GERARDAS ŽILINSKAS K,= min (max (2142) Ei+21/1 S Ri i D rg a DSS By the same argument as belore, the minimum is attained when 1 1 (22) T-Vi+2z giving A—31241=0. This gives p-31Y5 LžESĖ 2 2 and since …
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ON THE PRODUCT OF FOUR HEMOGENEOUS LINEAR FORMES 47 and ET 9, (x.y, 2) = 21 GE T EI] —l. Then for every set of values of x, y, z, 2 lx, y, z) = v4lx, y, Z); el y, 2) = 9, (x, y, z). Without loss of generality, we may suppose …
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48 GERARDAS ŽILINSKAS We consider three intervals for x; 4 > (a) x …
