On complete system of invariants for the binary form of degree 7

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Journal of Symbolc Compuaon 42 (2007 935 947 www.elsever.com/locae/jsc On complee sysem of nvarans for he bnary form of degree 7 Leond Bedrayuk Khmelnysky Naonal Unversy, Appled Mahemacs, Insyus ka s., Khmel nysky, 2906, Ukrane Receved 29 November 2006; acceped 25 July 2007 Avalable onlne 6 Augus 2007 Absrac A mnmal sysem of homogeneous generang elemens of he algebra of nvarans for he bnary form of degree 7 s calculaed. c 2007 Elsever Ld. All rghs reserved. Keywords: Classcal nvaran heory; Invarans of bnary form; Dervaons. Inroducon Le V n be he vecor C-space of he bnary forms of degree d equpped wh he naural acon of he group G = SL(2, C. Consder he correspondng acon of he group G on he coordnae rngs C[V d ] and C[V d C 2 ]. Denoe by I d = C[V d ] G and by C d = C[V d C 2 ] G he subalgebras of G-nvaran polynomal funcons. In he language of classcal nvaran heory he algebras I d and C d are called he algebra of nvarans and he algebra of covarans for he bnary form of degree d, respecvely. Le I d + be an deal of I d generaed by all homogeneous elemens of posve degree. Denoe by Ī d a se of homogeneous elemens of I d + such ha her mages n I d + /(I d + 2 form a bass of he vecor space. The se Ī d s called he complee sysem of nvarans for he bnary form of degree d. Denoe by n d he cardnaly of he se Ī d. The complee sysem C d of covarans for he bnary form may be smlarly defned. The complee sysems of nvarans and covarans were a major objec of research n classcal nvaran heory n he 9h cenury. I can be readly shown ha n = 0, n 2 = and n 3 =. Tel.: +380 3822 49043. E-mal address: bedrayuk@ef.up.km.ua. 0747-77/$ - see fron maer c 2007 Elsever Ld. All rghs reserved. do:0.06/j.jsc.2007.07.003

936 L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 The complee sysems of nvarans and covarans n he case d = 4 were calculaed by Boole, Cayley, Esensen. The case d = 5 was calculaed by he effors of Cayley and Herme, see Dxmer (990. In parcular, hey showed ha n 4 = 2, n 5 = 4. The complee sysems of nvarans and covarans n he case d = 6 were calculaed by Gordan, see Gordan (885. He showed ha n 6 = 5. A complee sysem of 9 nvarans n he case d = 8 was compued by Gall (888 and Shoda (967. The case d = 7 was consdered by Gall (880. Gall esablshed ha n 7 33. Almos a cenury laer Dxmer and Lazard (986 found ha n 7 = 30. However, he queson: wha elemens, exacly, form he complee sysem of nvarans, has remaned open o hs day, see survey Dxmer (990. Gall found a non-mnmal se of 33 generaors of nvarans for bnary forms of degree 7. The symbolc presenaon of nvarans, whch Gall used, makes very dffcul o decde whch are rreducble. The curren paper proposes a dfferen symbolc presenaon of nvarans and presens for he frs me a mnmal se of 30 generang nvarans. To solve he compuaon problem we offer a represenaon of he nvarans, whch s an nermedae beween he hghly unweldy explc represenaon and he hghly compressed symbolc represenaon. Also, we fnd a new symbolc represenaon of he fundamenal nvarans, whch s dfferen from Gall s represenaon. The represenaon uses ransvecans of low orders. The frs sep of he smplfcaon s calculaon of sem-nvarans nsead of covarans. Now, consder a covaran as a polynomal of generang funcons of he polynomal algebra C[V d C 2 ]. Then, a sem-nvaran s jus he leadng coeffcen of a polynomal wh respec o usual lexcographcal orderng. A sem-nvaran s an nvaran of upper unpoen marx subalgebra of Le algebra sl 2. Le us denfy he algebra C[V d ] wh he algebra C[X d ] := C[, x, x 2,..., x d ]. Also, denfy he algebra C[V d C 2 ] wh he polynomal algebra C[, x, x 2,..., x d, Y, Y 2 ]. The generang elemens dervaons ( 0 0 0, ( 0 0 0 D := + 2x + + dx d, x x 2 x d D 2 := dx + (d x 2 + + x d. x x d of he angen Le algebra sl 2 ac on C[V d ] by I follows ha he algebra I d concdes wh he algebra of polynomal soluons of he followng frs order PDE sysem, see Hlber (993, Glenn (95: u u u + 2x + + nx d = 0, x x 2 x d u dx + (d x u u 2 + + x d = 0, x x d.e. I d = C[X d ] D C[X d ] D 2, where u C[X d ], and C[X d ] D := { f C[X d ] D ( f = 0}, =, 2. For he compuaon of sem-nvarans we nroduce he sem-ransvecan ha s an analogue of he ransvecan. In he curren paper, we fnd an effecve formula for compuaon of he sem-ransvecans. For he case d = 7, we calculae all rreducble sem-nvarans up o 3h degree. An nvaran of even degree n may be consdered as a sem-ransvecan of he form (*

L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 937 [u, v], where u, v are sem-nvarans of degree n 2. Then, usng he obaned sem-nvarans we compue all rreducble nvarans up o 26h degree. The nvaran of degree 30 was aken from Gall s paper, see Gall (880. We compue he nvaran and check ha s rreducble. In hs way, here we explcly compue a complee sysem of 30 nvarans of he bnary form of degree 7. All calculaons were done wh Maple. 2. Prelmnares To begn wh, we afford a smplfcaon of he represenaon of covarans. Le ϰ : C d C[X d ] D be he C-lnear map ha akes each homogeneous covaran of order k o s leadng coeffcen,.e. o a coeffcen of Y k. In he ermnology of he classcal nvaran heory, an elemen of he algebra C[X d ] D s called a sem-nvaran, he degree of a homogeneous covaran wh respec o he varables se X d s called he degree of he covaran and s degree wh respec o he varables se Y, Y 2 s called he order of he covaran. Suppose ha F = m ( f m Y m Y2 s a covaran of order m, ϰ(f = f 0 C[X d ] D. The classcal Rober s heorem, Robers (86, saes ha he covaran F s compleely and unquely deermned by s leadng coeffcen f 0, namely, F = m D2 ( f 0 Y m! Y2. On he oher hand, every sem-nvaran s a leadng coeffcen of some covaran, Glenn (95. Ths gve us a well-defned explc form of he nverse map namely, ϰ : C[X] D C d, ϰ (a = ord(a D 2 (a! Y ord(a Y 2, where a C[X] D and ord(a s he order of he elemen a wh respec o he locally nlpoen dervaon D 2,.e. ord(a = max{s, D2 s (a 0}. For nsance, snce ord( = d, we have ϰ ( = ord( D 2 (! d ( Y ord( d Y2 = Y d + x Y d Y2. As we see ϰ ( s jus he basc bnary form. From polynomal funcon pon of vew he covaran ϰ ( s he evaluaon map. I s clear ha an nvaran s a sem-nvaran of order zero. Thus, he problem of fndng a complee sysem of he algebra C d s equvalen o he problem of fndng a complee sysem of sem-covarans of he algebra C[X] D. On he oher hand, he problem of fndng a complee sysem of he algebra Ī d s equvalen o he problem of fndng a subsysem n C d such ha s generaed by elemens of order zero, hese beng well-known classcal resuls. =

938 L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 A srucure of algebras of consans for such locally nlpoen dervaons can be easly deermned, see, for example, van den Essen (2000. In parcular, for he dervaon D we ge C[X d ] D = C[, σ (x 2,..., σ (x d ] [ ] C[X d ], where σ : C[X d ] C(X d D s a rng homomorphsm defned by σ (a = D (aλ!, λ = x, a C[X d]. Afer an uncomplcaed smplfcaon, we oban σ (x = z +, where z C[X d ] D and 2 ( z := ( k x k x k k k + ( ( + x, = 2,..., d. In parcular, k=0 z 2 = x 2 x 2, z 3 = x 3 2 + 2x 3 3x x 2, z 4 = x 4 3 3x 4 + 6x 2 x 2 4x x 3 2, z 5 = x 5 4 + 4x 5 0x 3 x 2 + 0x 2 x 3 2 5x x 4 3, z 6 = x 6 5 5x 6 + 5x 4 x 2 20x 3 x 3 2 + 5x 2 x 4 3 6x x 5 4, z 7 = x 7 6 + 6x 7 2x 5 x 2 + 35x 4 x 3 2 35x 3 x 4 3 + 2x 2 x 5 4 7x x 6 5. Thus, we oban C[X d ] D = C[, z 2,..., z d ] [ ] C[X d ]. Hence, he generang elemens of he algebra C[X d ] D may be regarded as he fracon f (z 2,...,z d s, f C[Z d ] := C[, z 2,..., z d ], s Z +. To make a calculaon wh nvarans n such a represenaon we should specfy an acon of he operaor D 2 n he new[ coordnaes ], z 2,..., z d. Denoe by D he exenson of he dervaon D 2 o he algebra C[Z d ] : D := D 2 ( + D 2(z 2 z 2 + + D 2 (z d z d. In Bedrayuk (2006 we showed ha D( = nλ, D(σ (x 2 = (n 2σ (x 3 (n 4σ (x 2 λ, D(σ (x = (n σ (x + (n 2σ (x λ (n σ (x 2σ (x, for > 2.

L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 939 Takng no accoun σ (x = z +, λ = x we can oban an expresson for D(z, = 2,..., d. In parcular, for d = 7 we ge : D = 7x ( 5x z 3 + 8z 2 2 4z 4 + (20x z 4 24z 2 z 3 + 3z 5 z 3 z 4 + (2z 6 + 25x z 5 30z 2 z 4 + (z 7 + 30x z 6 36z 2 z 5 z 5 z 6 + 7(5x z 7 6z 2 z 6 + 5(2x z 2 + z 3. z 7 z 2 To calculae sem-nvarans we should have an analogue of he ransvecans. Suppose F = m ( m f Y m k ( k Y2, G = g are wo covarans of degrees m and k respecvely. Le r ( r (F, G r = ( r F r G Y r Y2 Y, r Y2 Y k Y 2, f, g C[Z d ] [ ], be her r h ransvecan (Hlber, 993; Glenn, 95. The followng lemma gves us he rule how o fnd he sem-nvaran ϰ((f, G r whou drec compung of he covaran (F, G r. Lemma. The leadng coeffcen ϰ((f, G r of he covaran (F, G r, 0 r mn(m, k s calculaed by he formula r ( r D ϰ((f, G r = ( (ϰ(f x D r (ϰ(g x [m] =0,...,x r =0 [k], r =0,...,x r =0 where [a] := a(a... (a (, a Z. Proof. In Hlber (993, p. 87, one may fnd ha r ( r ϰ((f, G r = ( f g r. m ( m f By comparng wo dfferen forms of he covaran F, namely m F = Y m Y2 D2 and F = ( f 0 ( we ge f m D = 2 ( f 0!. Thus, f = D 2 ( f 0 [m] r ϰ((f, G r = ( ( r D (ϰ(f [m]! = D (ϰ(f [m] D r (ϰ(g [k] r. Y m Y 2,. Smlarly g = D (ϰ(g [k]. Therefore, The dervaon D s such ha D(C[Z d ][ ] C[Z d, x ][ ]. Furher, we have [ ] [ ] D (C[Z 2 d ] C[Z d, x, x 2 ],

940 L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 and for allowed r we ge [ ] [ ] D (C[Z r d ] C[Z d, x, x 2,..., x r ]. Therefore, ϰ((f, G r C[Z d, x, x 2,..., x r ] [ ]. On he oher hand, snce ϰ((f, G r s always a sem-nvaran, we see ha he ncluson ϰ((f, G r C[Z d ][ ] s correc. Thus, n he expresson for ϰ((f, Gr afer cancellaon, all coeffcens of x, x 2,..., x r mus be equal o zero. Hence, ϰ((f, G r = ϰ((f, G r = = x =0,...,x r =0 r ( r D ( (ϰ(f [m] r ( r D ( (ϰ(f x [m] =0,...,x r =0 D r (ϰ(g [k] r x =0,...,x r =0 D r (ϰ(g [k] r x =0,...,x r =0. Le f, g be wo sem-nvarans. Ther numeraors are polynomals of z 2,..., z n wh raonal coeffcens. Then, he sem-nvaran ϰ((ϰ ( f, ϰ (g s a fracon and s numeraor s a polynomal of z 2,..., z n wh raonal coeffcens. Therefore, we may mulply ϰ((ϰ ( f, ϰ (g r by some raonal number q r ( f, g Q such ha he numeraor of he expresson q r ( f, gϰ((ϰ ( f, ϰ (g r s hen a polynomal wh an neger coprme coeffcens. Pu [ f, g] r := q r ( f, gϰ((ϰ ( f, ϰ (g r, 0 r mn(ord( f, ord(g. The expresson [ f, g] r s sad o be he rh sem-ransvecan of he sem-nvarans f and g. The followng saemens are drec consequences of correspondng properes of ransvecans, see Glenn (95: Lemma 2. Le f, g be wo sem-nvarans. Then, he followng condons hold: ( he sem-ransvecan [, f g] s reducble for 0 mn(d, max(ord( f, ord(g; ( f ord( f = 0, hen, [, f g] = f [, g] ; ( ord([ f, g] = ord( f + ord(g 2; (v ord(z 2 z 3 3 z d d = d( 2 + 3 + + d 2(2 2 + 3 3 + + d d. Le us consder an example. Suppose d = 7, f = z 2, g = z 3. Then, ord(z 2 = 7 2 2 2 = 0, and ord(z 3 = 5. We have D(z 2 = 5(2x z 2 + z 3, [0] 0 =, [0] =, D 2 (z 2 = 0(3x 2 z 2 + 9x z 3 + 6x 2 z 2 9z 2 2 + 2z 4 2, [0] 2 = 0, D(z 3 = 5x z 3 8z 2 2 + 4z 4, [5] 0 =, [5] =, D 2 (z 3 = 2(60x 2 z 3 252x z 2 2 + 56x z 4 + 45x 2 z 3 38z 2 z 3 + 6z 5 2, [5] 2 = 5,

and We ge L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 94 D(z 2 = 5(z 3 x =0,x 2 =0, D 2 (z 2 = 0( 9z 2 2 +2z 4, x =0,x 2 =0 2 D(z 3 = 8z 2 2 +4z 4 x =0,x 2 =0, D 2 (z 3 = 2( 38z 2z 3 +6z 5 x =0,x 2 =0 ϰ((ϰ (z 2, ϰ (z 3 2 = Afer smplfcaon we oban r=0 2. 2 ( 2 d ( r r (z 2 x d 2 r (z 3 x r [0] r =0,x 2 =0 [5]. 2 r =0,x 2 =0 ϰ((ϰ (z 2, ϰ (z 3 2 = 2 3z 2 2 z 3 9z 2 z 5 + 7z 3 z 4 35 2. I s obvously ha q 2 (z 2, z 3 = 35 2. Therefore, [z 2, z 3 ] 2 = q 2 (z 2, z 3 ϰ((ϰ (z 2, ϰ (z 3 2 = 3z 2 2 z 3 9z 2 z 5 + 7z 3 z 4 2 = 3x 3 x 4 + 6x 4 x 3 + 9x 2 x 5 2 + 7x 3 3 x 4 + 30x 3 x 2 x 2 + 24x x 2 2 x 4 28x 3 2 2 x 2x 3 x 2 2 + 3x 2 2 2 x 3 9x 2 3 x 5 9x 2 3 x. In he same way one may ge [, [z 2, z 3 ] 2 ] = 27z 2 4 78z 2 2 z 4 4z 4 2 + 69z 2 z 3 2 + 2z 3 z 5 + 9z 2 z 6 2 = 78x 2 2 3 x 4 26x 2 2 x 4 + 45x 2 3 x 2 + 2x 3 4 x 5 + 69x 2 3 x 3 2 73x 2 x 3 2 2 + 68x 5 x 3 249x 4 x 4 9x 2 x 6 3 + 9x 2 4 x 6 + 78x 3 x 5 2 + 27x 2 4 2 4x 4 2 4 02x 2 2 2 x x 3 + 303x 2 x 2 2 x 4 + 78x 3 x 2 x 3 + 52x 4 3 x x 3 90x x 2 3 x 5. I can be seen from hese examples ha a represenaon of he sem-nvarans as elemens of he algebra C[Z d ][ ] s more compac han her sandard represenaon as elemens of he algebra C[X d ]. A very rough emprcal esmae would be ha a sem-nvaran ϰ (F has erms n ([deg(f/d] + 2(ord(F + mes less han he correspondng covaran F. Moreover, from compung pon of vew, he sem-ransvecan formula s more effecve han he ransvecan formula. These wo favorable crcumsances, coupled wh ulzng he grea praccal powers of Maple sofware, allow us o compue a complee sysem of nvarans for he bnary form of degree 7. 3. Compuaon of auxlary sem-nvarans Pror o any calculaon of nvarans, we fnd all rreducble covarans up o 3h degree. To do so, we use an analogue of well-known, Glenn (95, Ω-process. Le C 7, be a subse of elemens of ϰ(c 7 whose degree s. We are seekng he elemens of he se C 7,+ as rreducble elemens of a bass of he vecor space generaed by sem-ransvecans of he form [, uv] r, u C 7,l, v C 7,k, l + k =, max(ord(u, ord(v r 7. I s a sandard lnear algebra problem. The unque sem-nvaran of degree one obvously s, ord( = 7. The sem-ransvecans [, ], =... 7 are equal o zero for odd. Pu

942 L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 dv := [, ] 4 = 3z 2 2 +z 4 = x 2 4 4x x 3 + 3x 2 2, ord(dv =6, dv 2 := [, ] 6 = z 6+5z 2 z 4 0z 3, 2 = x 4 6 6x x 5 + 5x 2 x 4 0x 2 3, ord(dv 2 =2, dv 3 := [, ] 2 = z 2 = x 2 x 2, ord(dv 2 =0. The polynomals 2, dv, dv 2, dv 3 are lnearly ndependen. Therefore, he se C 7,2 consss of he rreducble sem-nvarans dv, dv 2, dv 3 of 2nd degree. To defne he se C 7,3 consder he followng polynomals 3, dv, dv 2, dv 3, [, dv ], =... 6, [, dv 2 ], =, 2 [, dv 3 ], =... 7. By drec calculaon we selec sx rreducble sem-nvarans of degree 3: r = [, dv ] 4, ord(r = 5, r 2 = [, dv 3 ], ord(r 2 = 5, r 3 = [, dv 3 ] 3, ord(r 3 =, r 4 = [, dv 3 ] 4, ord(r 4 = 9, r 5 = [, dv 3 ] 5, ord(r 5 = 7, r 6 = [, dv 3 ] 7, ord(r 6 = 3. In he same way, by usng Lemma 2, we may calculae all ses C 7,, 3. Le us presen he lss of he generang elemens. The se C 7,4 consss of he followng 8 rreducble sem-nvarans: ch = [, r 5 ] 7, ord(ch = 0, ch 2 = [, r 3 ] 7, ord(ch 2 = 4, ch 3 = [, r 3 ] 2, ord(ch 3 = 4, ch 4 = [, r 3 ] 4, ord(ch 4 = 0, ch 5 = [, r 3 ] 5, ord(r 5 = 8, ch 6 = [, r ] 2, ord(r 6 = 8, ch 7 = [, r ] 3, ord(ch 7 = 6, ch 8 = [, r ] 4, ord(ch 6 = 4. The se C 7,5 consss of he followng 0 rreducble sem-nvarans: p = [, ch 6 ] 5, ord(p = 5, p 2 = [, ch 6 ] 6, ord(p 2 = 3, p 3 = [, ch 7 ] 2, ord(p 3 = 9, p 4 = [, ch 7 ] 3, ord(p 4 = 7, p 5 = [, ch 7 ] 5, ord(p 5 = 3, p 6 = [, ch 6 ] 3, ord(p 6 = 9, p 7 = [, ch 4 ] 2, ord(p 7 = 3, p 8 = [, ch 4 ] 5, ord(p 6 = 7, p 9 = [, dv 2]7, ord(p 9 = 5, p 0 = [, dv dv 2 ] 7, ord(p 0 =. The se C 7,6 consss of he followng 0 rreducble sem-nvarans: sh = [, p 5 ] 5, ord(sh = 6, sh 2 = [, p 7 ] 6, ord(sh 2 = 8, sh 3 = [, p 4 ] 5, ord(sh 3 = 4, sh 4 = [, p 4 ] 6, ord(sh 4 = 2, sh 5 = [, p 3 ] 2, ord(sh 5 = 2, sh 6 = [, p 3 ] 4, ord(sh 6 = 8, sh 7 = [, p 4 ] 4, ord(sh 7 = 6, sh 8 = [, r dv ] 7, ord(sh 6 = 4, sh 9 = [, r dv 2 ] 6, ord(sh 9 = 2, sh 0 = [, r 6 dv ] 7, ord(sh 0 = 2. The se C 7,7 consss of he followng 2 rreducble sem-nvarans: s = [, sh 5 ] 4, ord(s =, s 2 = [, sh 7 ] 4, ord(s 2 = 5, s 3 = [, r 2]7, ord(s 3 = 3, s 4 = [, sh ] 3, ord(s 4 = 7, s 5 = [, ch 7 dv ] 7, ord(s 5 = 5, s 6 = [, ch 7 dv 2 ] 7, ord(s 6 =, s 7 = [, r6 2]4, ord(s 7 = 5, s 8 = [, r6 2]6, ord(s 6 =, s 9 = [, r 6 r ] 6, ord(s 9 = 3, s 0 = [, r 6 r ] 7, ord(s 0 =, s = [, r 2]6, ord(s = 5, s 2 = [, sh 0, ord(s 2 = 7.

L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 943 The se C 7,8 consss of he followng 3 rreducble sem-nvarans: v = [, s 7 ] 3, ord(v = 6, v 2 = [, s 7 ] 4, ord(v 2 = 4, v 3 = [, ch 8 r 6 ] 7, ord(v 3 = 0, v 4 = [, ch 8 r ] 6, ord(v 4 = 4, v 5 = [, ch 8 r ] 7, ord(v 5 = 2, v 6 = [, ch 7 r 6 ] 7, ord(v 6 = 2, v 7 = [, ch 7 r ] 7, ord(v 7 = 4, v 8 = [, ch 8 r 6 ] 6, ord(v 6 = 2, v 9 = [, r 6 dv2 2]7, ord(v 9 = 0, v 0 = [, s 4 ] 2, ord(v 0 = 0, v = [, s 2] 4, ord(v = 6, v 2 = [, s ] 3, ord(v 2 = 6, v 3 = [, p 9 dv 2 ] 7, ord(v 3 = 0. The se C 7,9 consss of he followng rreducble sem-nvarans: de = [, sh 3 dv ] 7, ord(de = 3, de 2 = [, ch 7 ch 8 ] 7, ord(de 2 = 3, de 3 = [, p 5 r 6 ] 5, ord(de 3 = 3, de 4 = [, p 5 r ] 6, ord(de 4 = 3, de 5 = [, p 5 r ] 7, ord(de 5 =, de 6 = [, sh 9 dv ] 7, ord(de 6 =, de 7 = [, sh 0 dv ] 7, ord(de 7 =, de 8 = [, sh 0 dv 2 ] 3, ord(de 6 = 5, de 9 = [, v 5 ] 2, ord(de 9 = 5, de 0 = [, v 2 ] 4, ord(de 0 = 3, de = [, v ] 2, ord(de = 9. The se C 7,0 consss of he followng 9 rreducble sem-nvarans: des = [, sh 9 r ] 6, ord(des = 2, des 2 = [, sh 4 r 6 ] 4, ord(des 2 = 4, des 3 = [, sh 4 r ] 6, ord(des 3 = 2, des 4 = [, sh r ] 7, ord(des 4 = 4, des 5 = [, sh 3 r 6 ] 5, ord(des 5 = 4, des 6 = [, de 9 ] 2, ord(des 6 = 8, des 7 = [, r6 3]7, ord(des 7 = 2, des 8 = [, sh 0 r ] 6, ord(des 6 = 2, des 9 = [, p ch 7 ] 7, ord(des 9 = 4. The se C 7, consss of he followng 9 rreducble sem-nvarans: odn = [, v 2 dv ] 7, ord(odn = 3, odn 2 = [, v 2, dv 2 ] 6, ord(odn 2 =, odn 3 = [, v 4 dv 2 ] 6, ord(odn 3 =, odn 4 = [, v 5 dv ] 7, ord(odn 4 =, odn 5 = [, v 6 dv ] 7, ord(odn 5 =, odn 6 = [, v 2 dv 2 ] 5, ord(odn 6 = 3, odn 7 = [, des 6 ] 4, ord(odn 7 = 7, odn 8 = [, des 6 ] 6, ord(odn 6 = 3, odn 9 = [, v dv 2 ] 7, ord(odn 9 =. The se C 7,2 consss of he followng 3 rreducble sem-nvarans: dvan = [, sh p 2 ] 7, ord(dvan = 2, dvan 2 = [, sh p 5 ] 7, ord(dvan 2 = 2, dvan 3 = [sh 9, sh 0 ] 2, ord(dvan 3 = 0, dvan 4 = [, odn 7 ] 6, ord(dvan 4 = 2, dvan 5 = [, de 8 dv 2 ] 6, ord(dvan 5 = 2, dvan 6 = [sh 0, sh 0 ] 2, ord(dvan 6 = 0, dvan 7 = [, de 9 dv 2 ] 6, ord(dvan 7 = 2, dvan 8 = [, de 0 dv ] 7, ord(dvan 6 = 2, dvan 9 = [, odn 7 ] 4, ord(dvan 9 = 6, dvan 0 = [sh, sh ] 2, ord(dvan 0 = 0, dvan = [sh 4, sh 4 ] 2, ord(dvan = 0, dvan 2 = [sh 4, sh 9 ] 2, ord(dvan 2 = 0, dvan 3 = [sh 4, sh 2 ] 2, ord(dvan 3 = 0. The se C 7,3 consss of he followng 9 rreducble sem-nvarans: ryn = [, dvan 9 ] 6, ord(r yn =, ryn 2 = [, v ch 7 ] 7, ord(ryn 2 = 5, ryn 3 = [, v 2 ch 8 ] 7, ord(r yn 3 =, ryn 4 = [, v 2 ch 2 ] 7, ord(ryn 4 =, ryn 5 = [, v ch 8 ] 7, ord(r yn 5 = 3, ryn 6 = [, v 5 ch 2 ] 6, ord(ryn 6 =, ryn 7 = [, v 8 ch 8 ] 6, ord(r yn 7 =, ryn 8 = [, v 8 ch 7 ] 7, ord(ryn 6 =, ryn 9 = [, v 4 ch 8 ] 7, ord(r yn 9 =.

944 L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 A number of elemens of C 7,, =,..., 3 and orders of he elemens so far concde compleely wh Gall s resuls, see Gall (888. 4. Compuaon of he nvarans Pu I := I 7 C 7,, I + := I 7 +. Le (I + 2 be a subse of (I 7 + 2, whose elemens have degree. Denoe by δ he number of rreducble nvarans of degree. I s clear ha δ = dm I dm(i+ 2, see Dxmer and Lazard (986. The dmenson of he vecor space I s calculaed by Cayley Sylveser formula, see, for example, Hlber (993 and Sprnger (977. The dmenson of he vecor space (I+ 2 s calculaed by he formula dm(i+ 2 = σ dm S. Here σ s he coeffcen of x n he seres expanson ( k< ( xk δ k, and S s a vecor subspace of (I 2 + generaed by syzyges. The nvarans of 4h, 8h and 2h degrees were found above. We have I 4 = p 4, δ 4 =, p 4 := ch = [, r 5 ] 7, I 8 = p 8,, p 8,2, p 8,3, δ 8 = 3, p 8, := v 3 = [, ch 8 r 6 ] 7, p 8,2 := v 9 = [, r 6 dv 2 2 ]7, p 8,3 := v 3 = [, p 9 dv 2 ] 7, I 2 = p 2,, p 2,2, p 2,3, p 2,4, p 2,5, p 2,6, δ 2 = 6, p 2, := dvan 3 = [sh 9, sh 0 ] 2, p 2,2 := dvan 6 = [sh 0 sh 0 ] 2, p 2,3 := dvan 0 = [sh sh ] 2, p 2,4 := dvan = [sh 4 sh 4 ] 2, p 2,5 := dvan 2 = [sh 4, sh 9 ] 2, p 2,6 := dvan 3 = [sh 4 sh 2 ] 2. For I 4 we have dm I 4 = 4, σ 4 = 0. Therefore (I 2 + 4 = 0, hen, δ 4 = 4. I s enough o fnd 4 lnearly ndependen nvarans of degree 4. Gven below s a ypcal nsance of how hese nvarans are calculaed. We are searchng he Invarans of I 4 as sem-ransvecans of he form [u, v], where u, v are sem-nvarans of C 7,7, < 4, ord(u + ord(v 2 = 0. There are 9 such sem-ransvecans: [s 8, s 8 ], [s 8, s 0 ], [s 9, s 9 ] 3, [s 0, s 0 ], [s 6, s 6 ], [s 6, s 8 ], [s 6, s 0 ], [s 3, s 3 ] 3, [s 3, s 9 ] 3. Usng Maple calculaon we choose four lnearly ndependen nvarans: p 4, := [s 8, s 0 ], p 4,2 := [s 6, s 0 ], p 4,3 := [s 6, s 8 ], p 4,4 := [s 3, s 9 ] 3. The nvarans p 4,, p 4,2, p 4,3, p 4,4 are fracons wh he denomnaor 35. The numeraors of he fracons are polynomals of Z[z 2,..., z 7 ] whch conss of 937, 869, 978, 925 erms respecvely. For I 6 we have dm I 6 = 8, σ 6 = 6. The vecor space (I 2 + 6 s generaed by 6 elemens and all of hem are lnearly ndependen. Thus δ 6 = 2. In order o calculae he nvarans of I 6, consder a se of sem-ransvecans of he form [u, v], where u, v C 7,8, < 5. There are 2 such sem-ransvecans: [v 7, v 7 ] 4, [v 5, v 5 ] 2, [v 5, v 6 ] 2, [v 5, v 8 ] 2, [v 6, v 6 ] 2, [v 6, v 8 ] 2, v 2, v 7 ] 4, [v 4, v 7 ] 4, [v 2, v 2 ] 4, [v 2, v 4 ] 4, [v 8, v 8 ] 2, [v 4, v 4 ] 4. In order o separae wo lnearly ndependen sem-ransvecans consder he equaly: α p 4 4 + α 2 p 2 8 + + α 7[v 7, v 7 ] 4 + + α 28 [v 4, v 4 ] 4 = 0.

L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 945 Subsung he values of he nvarans no he equaly we oban an over-defned sysem of lnear equaons. Afer solvng he sysem we ge 8 elemens of whch 6 are bass elemens of he vecors space (I 2 + 6 and 2 nvarans p 6, := [v 2, v 4 ] 4, and p 6,2 := [v 4, v 7 ] 4 ha span he vecor space I 6. The nvarans p 6,, p 6,2 are fracons wh he denomnaor 40. The numeraors of he fracons are polynomals of Z[z 2,..., z 7 ] whch conss of 744 and 698 erms. For I 8 we have dm I 8 = 3, σ 8 = 4. Snce (I 2 + 8 = p 4 I 4, hen, S 8 = 0. Thus δ 8 = 9. The nvarans of I 8 we are searchng as sem-ransvecans of he form [u, v], where u, v are sem-nvarans of C 7,9, < 6, ord(u + ord(v 2 = 0. In he same way as above, we oban he nne rreducble nvarans: p 8, := [de 4, de 3 ] 3, p 8,2 := [de 4, de 0 ] 3, p 8,3 := [de 5, de 6 ], p 8,4 := [de, de 0 ] 3, p 8,5 := [de 2, de 3 ] 3, p 8,6 := [de 2, de 0 ] 3, p 8,7 := [de 3, de 0 ] 3, p 8,8 := [de 6, de 7 ], p 8,9 := [de 8, de 9 ] 5. The nvarans p 8,, p 8,2,..., p 8,9 are fracons wh he denomnaor 45. The numeraors of he fracons are polynomals of Z[z 2,..., z 7 ] whch conss of 2674, 2758, 2645, 2800, 278, 2772, 2769, 266, 2739 erms respecvely. For I 20 we have dm I 20 = 35, σ 20 = 36. The vecor space S 20 s spanned by he wo syzyges: 42725300p 2,4 p 4 2 5449224200p 8, p 8,2 p 4 + 320855600p 8,3 p 8, p 4 32728580p 8,3 p 8,2 p 4 75375000p 2,6 p 8, + 398920p 8,3 2 p 4 69200p 6,2 p 4 + 530000p 2,4 p 8,3 9980000p 2,5 p 8,2 +3075755000p 8, 2 p 4 +890487920p 8,2 2 p 4 6875000p 2,2 p 8, 2025000p 2, p 8,3 + 34290000p 2, p 8,2 5200000p 2, p 8, and + 360450000p 2,5 p 8, 0800000p 2,4 p 8,2 + 9700000p 2,4 p 8, 675000p 6, p 4 337500p 2,2 p 8,3 507500p 2,6 p 8,3 + 2970000p 2,5 p 8,3 88556795p 2,5 p 4 2 + 4050000p 2,2 p 8,2 + 9440000p 2,6 p 8,2 35229030p 8, p 4 3 + 24000p 2,2 p 4 2 + 35439332p 8,2 p 4 3 2797842p 8,3 p 4 3 + 0336985p 2, p 4 2 = 0, 44093300p 2,4 p 4 2 + 549944200p 8, p 8,2 p 4 328535600p 8,3 p 8, p 4 + 329009580p 8,3 p 8,2 p 4 68625000p 2,6 p 8, 408520p 8,3 2 p 4 0768985p 2, p 4 2 440000p 2,4 p 8,3 4940000p 2,5 p 8,2 3229355000p 8, 2 p 4 903639920p 8,2 2 p 4 + 6875000p 2,2 p 8, + 2025000p 2, p 8,3 + 270000p 2, p 8,2 + 675000p 6, p 4 + 676350000p 2,5 p 8, + 0800000p 2,4 p 8,2 0700000p 2,4 p 8, + 337500p 2,2 p 8,3 372500p 2,6 p 8,3 + 430000p 2,5 p 8,3 + 69200p 6,2 p 4 4050000p 2,2 p 8,2 + 520000p 2,6 p 8,2 + 3657030p 8, p 4 3 + 336000p 2,6 p 4 2 2600000p 2, p 8, 35682532p 8,2 p 4 3 + 287042p 8,3 p 4 3 + 86396795p 2,5 p 4 2 = 0. Hence dm S 20 = 2 and δ 20 =. Ths confrms he resul repored n he paper Dxmer and Lazard (986. We fnd he unque nvaran I 20 as sem-ransvecan of he form [u, v], where u, v C 7,0, < 3, ord(u + ord(v 2 = 0. As a resul of he calculaon we ge ha he elemen

946 L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 p 20 := [des 7, des 7 ] 2 s he rreducble nvaran of degree 20. The nvaran p 20 s a fracon wh he denomnaor 50. The numeraor of he fracon s a polynomal of Z[z 2,..., z 7 ] whch consss of 4392 erms. For I 22 we have dm I 22 = 26, σ 22 = 25. By drec calculaon we fnd ha hose 25 elemens of (I 2 + 22 sasfy he unque syzygy p 4,3 p 8,3 4p 4,2 p 8,3 + 40p 4,3 p 8, + 750p 4, p 8,3 + 50p 4,4 p 8,3 60p 4,2 p 8, 275p 4,4 p 8,2 2p 4,3 p 8,2 875p 8,8 p 4 + 84p 4,2 p 8,2 + 48750p 4, p 8, + 9750p 4,4 p 8, + 25p 4, p 8,2 = 0. Hence dm S 22 = and δ 22 = 26 25 + = 2. I concdes wh he resuls of Dxmer and Lazard (986. We fnd he nvarans of I 22 as sem-ransvecans of he form [u, v], where u, v C 7,, < 4, ord(u + ord(v 2 = 0. As resul of he calculaon we ge ha he elemens p 22, := [odn 6, odn ] 3, p 22,2 := [odn 8, odn ] 3 are rreducble nvarans of degree 22. The nvarans p 22,, p 22,2 are fracons wh he denomnaor 55. The numeraors of he fracons are polynomals of Z[z 2,..., z 7 ] whch conss of 6569 and 6556 erms respecvely. For I 24 we have dm I 24 = 62, σ 24 = 74. By drec calculaon we ge ha he vecor space S 24 s spanned by 2 syzyges. Hence δ 24 = 62 74 + 2 = 0. For I 26 we have dm I 26 = 52, σ 26 = 78. By drec Maple calculaon we oban ha he vecor space S 26 s spanned by 27 syzyges. Hence, δ 26 = 52 78 + 27 =, ha concdes wh he resul of Dxmer and Lazard (986. Invarans of I 26 we are seekng as a sem-ransvecans of he form [u, v], where u, v C 7,3, =, ord(u + ord(v 2 = 0. As resul of he calculaon we ge ha he elemen p 26 = [ryn 4, r yn 3 ] s he unque rreducble nvaran of degree 26. The nvaran p 26 s a fracon wh he denomnaor 65. The numeraor of he fracon s a polynomal of Z[z 2,..., z 7 ] whch consss of 3 65 erms. For I 28 we have dm I 28 = 97, σ 28 = 35. By drec calculaon we ge he vecor space S 28 spanned by 38 syzyges. Hence δ 28 = 97 35 + 38 = 0. For I 30 we have dm I 30 = 92, σ 30 = 7. By drec Maple calculaon we oban ha he vecor space S 30 s spanned by 80 syzyges. Hence, δ 30 = 92 7 + 80 =. A unque rreducble nvaran of I 30 we ake from Gall s paper, Gall (888. In he paper s noaon we fnd ha p 30 = (h, α. Afer all calculaons we oban ha p 30 s a fracon wh he denomnaor 75. The numeraor of he fracon s a polynomal of Z[z 2,..., z 7 ] whch consss of 25 868 erms. Summarzng he above resuls we ge Theorem. The followng sysem of 30 nvarans s a complee sysem of he nvarans for he bnary form of degree 7

L. Bedrayuk / Journal of Symbolc Compuaon 42 (2007 935 947 947 p 4 = [, r 5 ] 7, p 8, = [, ch 8 r 6 ] 7, p 8,2 = [, r 6 dv2 2 ]7, p 8,3 = [, p 9 dv 2 ] 7, p 2, = [sh 9, sh 0 ] 2, p 2,2 = [sh 0 sh 0 ] 2, p 2,3 = [sh sh ] 2, p 2,4 = [sh 4 sh 4 ] 2, p 2,5 = [sh 4, sh 9 ] 2, p 2,6 = [sh 4 sh 2 ] 2, p 4, = [s 8, s 0 ], p 4,2 = [s 6, s 0 ], p 4,3 = [s 6, s 8 ], p 4,4 = [s 3, s 9 ] 3, p 6, = [v 2, v 4 ] 4, p 6,2 = [v 4, v 7 ] 4, p 8, = [de 4, de 3 ] 3, p 8,2 = [de 4, de 0 ] 3, p 8,3 = [de 5, de 6 ], p 8,4 = [de, de 0 ] 3, p 8,5 = [de 2, de 3 ] 3, p 8,6 = [de 2, de 0 ] 3, p 8,7 = [de 3, de 0 ] 3, p 8,8 = [de 6, de 7 ], p 8,9 = [de 8, de 9 ] 5, p 20 = [des 7, des 7 ] 2, p 22, = [odn 6, odn ] 3, p 22,2 = [odn 8, odn ] 3, p 26 = [ryn 4, r yn 3 ], p 30 = (h, α (n Gall s noaon. Acknowledgemens The auhor s graeful o Dr. Ivan Arzhansev and Dr. Farogh Dovlashah for he useful dscussons. References Bedrayuk, L., 2006. On he dfferenal equaon of nvarans of bnary forms. mah.ag/0602373. Dxmer, J., 990. Quelques aspecs de la héore des nvarans. Gaz. Mah., Soc. Mah. Fr. 43, 39 64. Dxmer, J., Lazard, D., 986. Le nombre mnmum d nvarans fondamenaux pour les formes bnares de degré 7. Por. Mah. 43, 377 392. Gall, F., 880. Das vollsändg formensysem der bnären form acher ordnung. Mah. Ann. (7, 3 52, 39 52. Gall, F., 888. Das vollsändge formensysem der bnären form 7 er ordnung. Mah. Ann. (3, 38 336. Glenn, O.E., 95. Trease on heory of nvarans. Boson. Gordan, P., 885. Invaranenheore. Teubner, Lepzg (reprned by Chelsea Publ. Co., 987, Boson. Hlber, D., 993. Theory of algebrac nvarans. In: Lecures. Cambrdge Mahemacal Lbrary. Cambrdge Unversy Press, Cambrdge. Robers, M., 86. The covarans of a bnary quanc of he nh degree. Quarerly J. Mah. 4, 68 78. Shoda, T., 967. On he graded rng of nvarans of bnary ocavcs. Am. J. Mah. 89, 022 046. Sprnger, T., 977. Invaran heory. In: Lecure Noes n Mahemacs, vol. 585. Sprnger-Verlag, Berln, Hedelberg, New York. van den Essen, A., 2000. Polynomal Auomorphsms and he Jacoban Conjecure. In: Progress n Mahemacs (Boson, Mass, vol. 90, Basel.

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