Invariant Theory of Finite Groups by Maple 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2MFEifkYnLyUlc2l6ZUdRIzE2RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRktGK0YvLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RicvRjZRJ25vcm1hbEYn JSFH 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 Theoritical Background Let F be a field of characteristic 0 and F[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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= G acts on F[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 For A\342\210\210G and f\342\210\210F[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 (A.f)(X) = f(AX), where X=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A.f \342\210\210F[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 We have the following i) I.f = f ii) A.(B.f) = (AB).f, where I is the identity matrix and A,B \342\210\210G. A polynomial f(X) is called invariant under G if f(AX) = f(X) for any A\342\210\210G. F[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 Theorem 1 F[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F[LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EieEYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2KS1JI21uR0YkNiZRIjFGJ0YyRjhGO0YyLyUrZm9yZWdyb3VuZEdRLFsyMDAsMCwyMDBdRicvJSxwbGFjZWhvbGRlckdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRJVRleHRGJ0Y4RjsvJS9zdWJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi9RIixGJ0YyRjhGOy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjMzMzMzMzNlbUYnLUYsNiVGLi1GIzYpLUZANiZRIjJGJ0YyRjhGO0YyRkNGRkZIRjhGO0ZLRk4tRk82L1EjLi5GJ0YyRjhGO0ZSL0ZVRjdGVkZYRlpGZm5GaG4vRltvUSwwLjIyMjIyMjJlbUYnL0Zeb0Zcby1GTzYvUSIuRidGMkY4RjtGUkZqb0ZWRlhGWkZmbkZobkZqbkZdcEZOLUYsNiVGLi1GIzYpLUYvNidRIm5GJ0YyRjVGOEY7RjJGQ0ZGRkhGOEY7RkstRk82L1EiXUYnRjJGOEY7L0ZTRjRGam8vRldGNEZYRlpGZm5GaG4vRltvUSwwLjE2NjY2NjdlbUYnL0Zeb1EsMC4xMTExMTExZW1GJy1GTzYvUSJ+RidGMkY4RjtGUkZqb0ZWRlhGWkZmbkZobkZqbkZdcEZecEYyRkhGOEY7 Theorem 2 (Hilbert) 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Reynolds operator of G is the mapping 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F[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[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 Theorem 3 i) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYxLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiUkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2Jy1GLzYnUSJHRidGMkY1RjhGO0YyLyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjhGOy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2L1EifkYnRjJGOEY7LyUmZmVuY2VHRjcvJSpzZXBhcmF0b3JHRjcvJSlzdHJldGNoeUdGNy8lKnN5bW1ldHJpY0dGNy8lKGxhcmdlb3BHRjcvJS5tb3ZhYmxlbGltaXRzR0Y3LyUnYWNjZW50R0Y3LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGZm5GSC1GLzYnUSNpc0YnRjJGNUY4RjtGSC1GLzYnUSJhRidGMkY1RjhGO0ZILUYvNidRJ2xpbmVhckYnRjJGNUY4RjtGSC1GLzYnUShvcGVyYXRvRidGMkY1RjhGOy1GLzYnUSJyRidGMi9GNkY0L0Y5USxib2xkLWl0YWxpY0YnRjtGMkZob0Zpb0Y7 ii) LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUklbXN1YkdGJDYlLUkjbWlHRiQ2J1EiUkYnLyUlYm9sZEdRJXRydWVGJy8lJ2l0YWxpY0dRJmZhbHNlRicvJSxtYXRodmFyaWFudEdRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRjotRiM2Jy1GLzYnUSJHRidGMkY1RjhGO0YyLyUwZm9udF9zdHlsZV9uYW1lR1ElVGV4dEYnRjhGOy8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkobWZlbmNlZEdGJDYmLUYjNiYtRi82J1EiZkYnRjJGNUY4RjtGMkY4RjtGMkY4RjstSSNtb0dGJDYvUSomRWxlbWVudDtGJ0YyRjhGOy8lJmZlbmNlR0Y3LyUqc2VwYXJhdG9yR0Y3LyUpc3RyZXRjaHlHRjcvJSpzeW1tZXRyaWNHRjcvJShsYXJnZW9wR0Y3LyUubW92YWJsZWxpbWl0c0dGNy8lJ2FjY2VudEdGNy8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRl5vRjJGOEY7F[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 iii) 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F[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 Theorem 4 (Noether) F[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 Theorem 5 (Molien's Series) H(t) =LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzZbby1JI21vR0YkNi1RIn5GJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjQvJSlzdHJldGNoeUdGNC8lKnN5bW1ldHJpY0dGNC8lKGxhcmdlb3BHRjQvJS5tb3ZhYmxlbGltaXRzR0Y0LyUnYWNjZW50R0Y0LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdGQy1JJm1mcmFjR0YkNigtSSNtbkdGJDYmUSIxRicvJSVib2xkR1EldHJ1ZUYnL0YwUSVib2xkRicvJStmb250d2VpZ2h0R0ZRLUYjNiYtSShtZmVuY2VkR0YkNiYtRiM2Ji1JI21pR0YkNidRIkdGJ0ZNLyUnaXRhbGljR0Y0RlBGUkZNRlBGUkYvLyUlb3BlbkdRInxnckYnLyUmY2xvc2VHRl1vRk1GUEZSLyUubGluZXRoaWNrbmVzc0dGTC8lK2Rlbm9tYWxpZ25HUSdjZW50ZXJGJy8lKW51bWFsaWduR0Zkby8lKWJldmVsbGVkR0Y0LUkrbXVuZGVyb3ZlckdGJDYnLUYsNi9RJiZTdW07RidGTUZQRlIvRjNRJnVuc2V0RicvRjZGYHAvRjhGTy9GOkZgcC9GPEZPL0Y+Rk8vRkBGYHBGQS9GRVEsMC4xNjY2NjY3ZW1GJy1GIzYoLUZmbjYnUSJBRidGTUZpbkZQRlItRiw2L1EqJkVsZW1lbnQ7RidGTUZQRlJGMkY1RjdGOUY7Rj1GPy9GQlEsMC4yNzc3Nzc4ZW1GJy9GRUZicUZlbkZNRlBGUi1GZm42I1EhRidGPy8lLGFjY2VudHVuZGVyR0Y0LUZHNihGSS1GIzYnLUZmbjYnUSRkZXRGJ0ZNRmluRlBGUi1GVzYmLUYjNigtRmZuNidRIklGJ0ZNRmluRlBGUi1GLDYvUSgmbWludXM7RidGTUZQRlJGMkY1RjdGOUY7Rj1GPy9GQlEsMC4yMjIyMjIyZW1GJy9GRUZbcy1GZm42J1EjdEFGJ0ZNRmluRlBGUkZNRlBGUkZNRlBGUkZNRlBGUkZgb0Zib0Zlb0Znby1GLDYvRi5GTUZQRlJGMkY1RjdGOUY7Rj1GP0ZBRkQtRiw2L1EiPUYnRk1GUEZSRjJGNUY3RjlGO0Y9Rj9GYXFGY3FGYHNGYHMtSSVtc3ViR0YkNiUtRmZuNidRImFGJ0ZNL0ZqbkZPL0YwUSxib2xkLWl0YWxpY0YnRlItRiM2KS1GSjYmUSIwRidGTUZQRlJGTUZbdC8lK2ZvcmVncm91bmRHUSxbMjAwLDAsMjAwXUYnLyUscGxhY2Vob2xkZXJHRk9GXHRGUi8lL3N1YnNjcmlwdHNoaWZ0R0ZidC1GLDYvUSIrRidGTUZQRlJGMkY1RjdGOUY7Rj1GP0ZqckZccy1GZnM2JUZocy1GIzYpRklGTUZbdEZjdEZmdEZcdEZSRmh0RmBzLUZmbjYnUSJ0RidGTUZbdEZcdEZSRmp0LUYsNi9RIy4uRidGTUZQRlJGMkY1RjdGOUY7Rj1GP0ZqckZELUYsNi9RIi5GJ0ZNRlBGUkYyRjVGN0Y5RjtGPUY/RkFGREZqdC1GZnM2JUZocy1GIzYtLUYsNi1GXW9GL0YyRjVGYnBGOUY7Rj1GPy9GQlEsMC4xMTExMTExZW1GJy9GRUZhdi1GZm42J0ZobkZNRlt0Rlx0RlJGYHNGXnZGK0ZNRlt0RmN0RmZ0Rlx0RlJGaHQtSSVtc3VwR0YkNiVGYXUtRiM2Jy1GVzYmLUYjNidGY3ZGTUZbdEZcdEZSRi9GW29GXm9GTUZbdEZcdEZSLyUxc3VwZXJzY3JpcHRzaGlmdEdGYnRGYHNGanQtRmZuNidRIk9GJ0ZNRmluRlBGUi1GLDYtUSIoRidGLy9GM0ZPRjVGYnBGOUY7Rj1GPy9GQkZocEZncC1GZnY2JUZhdS1GIzYrRklGanRGXnZGY3ZGXnZGTUZbdEZcdEZSRl53LUYsNi1RIilGJ0YvRmZ3RjVGYnBGOUY7Rj1GP0Znd0ZncEZgcy1GLDYvUSIsRidGTUZQRlJGMi9GNkZPRjdGOUY7Rj1GP0ZBL0ZFUSwwLjMzMzMzMzNlbUYnRmBzRmBzRmBzLUZmbjYnUSZ3aGVyZUYnRk1GaW5GUEZSRmBzRmBzLUZmczYlLUZmbjYnRmpzRk1GaW5GUEZSLUYjNigtRmZuNidRImlGJ0ZNRmluRlBGUkZNRmN0RmZ0RlBGUkZodEZgc0Zgcy1GZm42J1EjaXNGJ0ZNRmluRlBGUkZgcy1GZm42J1EkdGhlRidGTUZpbkZQRlJGYHMtRmZuNidRJ251bWJlckYnRk1GaW5GUEZSRmBzLUZmbjYnUSNvZkYnRk1GaW5GUEZSRmBzLUZmbjYnUSlsaW5lYXJseUYnRk1GaW5GUEZSRmBzLUZmbjYnUSxpbmRlcGVuZGVudEYnRk1GaW5GUEZSRmBzLUZmbjYnUSxob21vZ2VuZW91c0YnRk1GaW5GUEZSRmBzLUZmbjYnUStpbnZhcmlhbnRzRidGTUZpbkZQRlJGYHNGanlGYHNGYHMtRmZuNidRJ2RlZ3JlZUYnRk1GaW5GUEZSRmBzRmBzRl55Rmd1Rk1GUEZS Theorem 6 Let 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F[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F[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 i) f\342\210\210F[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 ii) if f\342\210\210F[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 Theorem 7 Let 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Method 1) Use Molien's series to find the number of linearly independent generators at each degree of 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2) Apply Reynolds operator to list the invariants for degree \342\251\275 |G| . The procedure invs is doing that. 3) By inspection we eliminate the algebraically dependent generators. 4) The relations among the generators are found by using Groebner basis according to theorem 7. The procedure relations find out the required relations. 5) Procedures invpol1 and invpol2 can be used to i) check if a given polynomial is invariant. ii) express an invariant polynomial in terms of the generators. 6) The computaions done are restricted to nly variables x,y. Matrix Groups LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYmLUkjbWlHRiQ2JVEocmVzdGFydEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JI21vR0YkNi1RIjtGJy9GM1Enbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPS8lKnN5bW1ldHJpY0dGPS8lKGxhcmdlb3BHRj0vJS5tb3ZhYmxlbGltaXRzR0Y9LyUnYWNjZW50R0Y9LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJStleGVjdXRhYmxlR0Y9Rjk= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1Enbm9ybWFsRictSShtZmVuY2VkR0YkNiQtRiM2JS1GLDYlUS5MaW5lYXJBbGdlYnJhRidGL0YyLyUrZXhlY3V0YWJsZUdGMUYyRjItSSNtb0dGJDYtUSI6RidGMi8lJmZlbmNlR0YxLyUqc2VwYXJhdG9yR0YxLyUpc3RyZXRjaHlHRjEvJSpzeW1tZXRyaWNHRjEvJShsYXJnZW9wR0YxLyUubW92YWJsZWxpbWl0c0dGMS8lJ2FjY2VudEdGMS8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRlNGPUYy LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEld2l0aEYnLyUnaXRhbGljR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWNGJy1JKG1mZW5jZWRHRiQ2JC1GIzYkLUYsNiVRKUdyb2VibmVyRidGL0YyL0YzUSdub3JtYWxGJ0Y9LUkjbW9HRiQ2LVEiOkYnRj0vJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkUvJSlzdHJldGNoeUdGRS8lKnN5bW1ldHJpY0dGRS8lKGxhcmdlb3BHRkUvJS5tb3ZhYmxlbGltaXRzR0ZFLyUnYWNjZW50R0ZFLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGVC8lK2V4ZWN1dGFibGVHRkVGPQ== Cyclic groups of order n generated by a matrix A 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 = {I, A,..., 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 Rotation groups Rn = < 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 and unity root groups Un = < 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= cyc:=proc(A,n) [seq(A^r,r=0..n-1)]; end proc;

Zio2JEkiQUc2IkkibkdGJUYlRiVGJTcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkKUYkSSJyR0YlL0YtOyIiISwmRiYiIiJGMiEiIkYlRiVGJQ==

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= {I, A, 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 = {I, A,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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= metacyc:=proc(A,B,n) local L,K; L:=[seq(A^r,r=0..n-1)]; K:= [seq(MatrixMatrixMultiply(L[s],B),s=1..n)]; [L[],K[]]; end proc;

Zio2JUkiQUc2IkkiQkdGJUkibkdGJTYkSSJMR0YlSSJLR0YlRiVGJUMlPkYpNyMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQpRiRJInJHRiUvRjM7IiIhLCZGJyIiIkY4ISIiPkYqNyMtRi82JC1fSS5MaW5lYXJBbGdlYnJhRzYkRjBJKF9zeXNsaWJHRiVJNU1hdHJpeE1hdHJpeE11bHRpcGx5R0YlNiQmRik2I0kic0dGJUYmL0ZHO0Y4Ric3JCZGKUYlJkYqRiVGJUYlRiU=

JSFH Symmetric groups 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Klien's group V4=D2 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Dihedral group of order 6 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Isomorphic copy of D3 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Dihedral group of order 8 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JSFH Quaternions group 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Generator 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 The symbol letter t is being used (reserved) for the variable of the series. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= molien:=proc(G) local h; h:=(1/nops(G)) *add(1/(Determinant(G[1]-t*A)) ,A in G); h:=simplify(h); taylor(h,t=0,nops(G)+1); end proc;

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LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnbW9saWVuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEjUjFGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjtGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGPUZA

KylJInRHNiIiIiIiIiEiIiNGJS1JIk9HJSpwcm90ZWN0ZWRHNiNGJUYn

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JVEnbW9saWVuRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkobWZlbmNlZEdGJDYkLUYjNiUtRiw2JVEjUjJGJ0YvRjIvJStleGVjdXRhYmxlR1EmZmFsc2VGJy9GM1Enbm9ybWFsRidGQC1JI21vR0YkNi1RIjtGJ0ZALyUmZmVuY2VHRj8vJSpzZXBhcmF0b3JHRjEvJSlzdHJldGNoeUdGPy8lKnN5bW1ldHJpY0dGPy8lKGxhcmdlb3BHRj8vJS5tb3ZhYmxlbGltaXRzR0Y/LyUnYWNjZW50R0Y/LyUnbHNwYWNlR1EmMC4wZW1GJy8lJ3JzcGFjZUdRLDAuMjc3Nzc3OGVtRidGPUZA

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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= act:=proc(A,f) local a,b,c,d; a:=A[1][1]; b:=A[1][2]; c:=A[2][1]; d:=A[2][2]; simplify(subs([x=a*x+b*y,y=c*x+d*y],f)); end proc;

Zio2JEkiQUc2IkkiZkdGJTYmSSJhR0YlSSJiR0YlSSJjR0YlSSJkR0YlRiVGJUMnPkYoJiZGJDYjIiIiRjA+RikmRi82IyIiIz5GKiYmRiRGNEYwPkYrJkY4RjQtSSlzaW1wbGlmeUc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJTYjLUklc3Vic0dGPjYkNyQvSSJ4R0YlLCYqJkYoRjFGRkYxRjEqJkYpRjFJInlHRiVGMUYxL0ZKLCYqJkYqRjFGRkYxRjEqJkYrRjFGSkYxRjFGJkYlRiVGJQ==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Reynolds:=proc(G,p); simplify(1/nops(G) *add(act(A,p),A in G)); end proc;

Zio2JEkiR0c2IkkicEdGJUYlRiVGJS1JKXNpbXBsaWZ5RzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiMqJi1JJW5vcHNHRio2I0YkISIiLUkkYWRkR0YqNiQtSSRhY3RHRiU2JEkiQUdGJUYmLUkjaW5HRio2JEY4RiQiIiJGJUYlRiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRistSSNtaUdGJDYnUSptb25vbWlhbHNGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHRjQvRjBRJWJvbGRGJy8lK2ZvbnR3ZWlnaHRHRlAvJStleGVjdXRhYmxlR0ZMLyUwZm9udF9zdHlsZV9uYW1lR1EpMkR+SW5wdXRGJ0Yv LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= mon:=proc(x,y,n) local K,L; K:=[seq(x^(n-i),i=0..n)]; L:=[seq(y^j,j=0..n)]; [seq(K[r]*L[r],r=1..n+1)]; end proc;

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Zio2JUkieEc2IkkieUdGJUkibkdGJUYlRiVGJTcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkJi1JJG1vbkdGJTYlRiRGJkkic0dGJUYlL0YxOyIiIkYnRiVGJUYl

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LCYqJiMiIiIiIiNGJSlJInhHNiJGJkYlRiUqJkYkRiUpSSJ5R0YpRiZGJUYl

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

LCYqJilJInhHNiIiIiMiIiJJInlHRiZGKCEiIiomRiVGKClGKUYnRihGKg==

LCYqKCMiIiIiIiNGJSlJInhHNiJGJkYlSSJ5R0YpRiVGJSooRiRGJUYoRiUpRipGJkYlRiU=

LCgqJiMiIiMiIiQiIiIpSSJ4RzYiRiVGJ0YnKihGJEYnRilGJ0kieUdGKkYnRicqJkYkRicpRixGJUYnRic=

LCgqKCMiIiIiIiRGJUkieEc2IkYlSSJ5R0YoRiUhIiIqJkYkRiUpRiciIiNGJUYqKiZGJEYlKUYpRi1GJUYq

LCoqKCMiIiIiIiRGJSlJInhHNiJGJkYlKUkieUdGKSIiI0YlISIiKigjRiUiIidGJSlGKCIiJUYlRitGJUYtKihGJEYlKUYoRixGJSlGK0YmRiVGLSooRi9GJUYoRiUpRitGMkYlRi0=

LCoqKCMiIiYiIiQiIiIpSSJ4RzYiIiIlRidJInlHRipGJyEiIiooIyIjNUYmRicpRilGJkYnKUYsIiIjRidGLSooRi9GJylGKUYzRicpRixGJkYnRi0qKEYkRidGKUYnKUYsRitGJ0Yt

LCoqKCMiIiYiIiciIiIpSSJ4RzYiIiIlRidJInlHRipGJ0YnKigjRiUiIiRGJylGKUYvRicpRiwiIiNGJ0YnKihGLkYnKUYpRjJGJylGLEYvRidGJyooRiRGJ0YpRicpRixGK0YnRic=

LCwqJiMiIiMiIiQiIiIpSSJ4RzYiIiIlRidGJyooI0YrRiZGJylGKUYmRidJInlHRipGJ0YnKihGJUYnKUYpRiVGJylGL0YlRidGJyooRi1GJ0YpRicpRi9GJkYnRicqJkYkRicpRi9GK0YnRic=

LCwqJilJInhHNiIiIiMiIiIpSSJ5R0YmRidGKEYoKiYjRigiIiRGKClGJSIiJUYoRigqKCNGJ0YtRigpRiVGLUYoRipGKEYoKihGMUYoRiVGKClGKkYtRihGKComRixGKClGKkYvRihGKA==

LCwqKCMiIiMiIiQiIiIpSSJ4RzYiRiZGJ0kieUdGKkYnISIiKiYjRidGJkYnKUYpIiIlRidGLComKUYpRiVGJylGK0YlRidGLCooRiRGJ0YpRicpRitGJkYnRiwqJkYuRicpRitGMEYnRiw=

LDAqJiMiIiMiIiQiIiIpSSJ4RzYiIiInRidGJyooRiVGJylGKSIiJkYnSSJ5R0YqRidGJyooRi5GJylGKSIiJUYnKUYvRiVGJ0YnKigjIiM/RiZGJylGKUYmRicpRi9GJkYnRicqKEYuRicpRilGJUYnKUYvRjJGJ0YnKihGJUYnRilGJylGL0YuRidGJyomRiRGJylGL0YrRidGJw==

LDAqKCMiIiIiIiRGJSlJInhHNiJGJkYlKUkieUdGKUYmRiUhIiIqJkYkRiUpRigiIidGJUYsKiYpRigiIiZGJUYrRiVGLComKUYoIiIlRiUpRisiIiNGJUYsKiYpRihGN0YlKUYrRjVGJUYsKiZGKEYlKUYrRjJGJUYsKiZGJEYlKUYrRi9GJUYs

LDAqKCMiIiQiIiMiIiIpSSJ4RzYiIiIlRicpSSJ5R0YqRiZGJ0YnKiYjRidGJUYnKUYpIiInRidGJyomKUYpIiImRidGLUYnRicqKCNGK0YlRicpRilGJUYnKUYtRiVGJ0YnKihGJEYnKUYpRiZGJylGLUYrRidGJyomRilGJylGLUY0RidGJyomRi9GJylGLUYxRidGJw==

LDAqJilJInhHNiIiIiYiIiJJInlHRiZGKCEiIiomI0YoIiIkRigpRiUiIidGKEYqKigjRiciIiNGKClGJSIiJUYoKUYpRjJGKEYqKigjIiM1Ri1GKClGJUYtRigpRilGLUYoRioqKEYxRigpRiVGMkYoKUYpRjRGKEYqKiZGJUYoKUYpRidGKEYqKiZGLEYoKUYpRi9GKEYq

JSFH 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

LCYqJiMiIiIiIiNGJSlJInhHNiJGJkYlRiUqJkYkRiUpSSJ5R0YpRiZGJUYl

LCYqJiMiIiIiIiVGJSlJInhHNiIiIiRGJUYlKigjRipGJkYlRihGJSlJInlHRikiIiNGJSEiIg==

LCYqKCMiIiQiIiUiIiJJInhHNiJGJylJInlHRikiIiNGJ0YnKiYjRidGJkYnKUYoRiVGJyEiIg==

LCgqJiMiIiQiIikiIiIpSSJ4RzYiIiIlRidGJyooI0YlRitGJylGKSIiI0YnKUkieUdGKkYvRidGJyomRiRGJylGMUYrRidGJw==

LCgqJiMiIiYiIzsiIiIpSSJ4RzYiRiVGJ0YnKigjRiUiIilGJylGKSIiJEYnKUkieUdGKiIiI0YnISIiKigjIiM6RiZGJ0YpRicpRjEiIiVGJ0Yz

LCgqKCMiIioiIzsiIiJJInhHNiJGJylJInlHRikiIiVGJ0YnKiYjIiIkRiZGJylGKCIiJkYnISIiKigjRi8iIilGJylGKEYvRicpRisiIiNGJ0Yn

LCgqKCMiIiIiIiVGJSlJInhHNiIiIiNGJSlJInlHRilGKkYlRiUqJiNGJSIiKUYlKUYoRiZGJUYlKiZGLkYlKUYsRiZGJUYl

LCgqKCMiIiIiIilGJSlJInhHNiIiIiRGJSlJInlHRikiIiNGJUYlKigjRioiIztGJUYoRiUpRiwiIiVGJUYlKiYjRiVGMEYlKUYoIiImRiUhIiI=

LCoqJiMiIzYiI0siIiIpSSJ4RzYiIiInRidGJyooIyIjOkYmRicpRikiIiVGJylJInlHRioiIiNGJ0YnKigjIiNYRiZGJylGKUYzRicpRjJGMEYnRicqJiMiIipGJkYnKUYyRitGJ0Yn

LCoqJiMiIzYiI0siIiIpSSJ5RzYiIiInRidGJyomIyIiKkYmRicpSSJ4R0YqRitGJ0YnKigjIiNYRiZGJylGMCIiJUYnKUYpIiIjRidGJyooIyIjOkYmRicpRjBGN0YnKUYpRjVGJ0Yn

LCoqKCMiI0AiI0siIiIpSSJ4RzYiIiIjRicpSSJ5R0YqIiIlRidGJyooIyIiKkYmRicpRilGLkYnKUYtRitGJyEiIiomI0YnRiZGJylGLSIiJ0YnRicqJiMiIiRGJkYnKUYpRjhGJ0Yn

LCoqKCMiI0AiI0siIiIpSSJ4RzYiIiIlRicpSSJ5R0YqIiIjRidGJyooIyIiKkYmRicpRilGLkYnKUYtRitGJyEiIiomI0YnRiZGJylGKSIiJ0YnRicqJiMiIiRGJkYnKUYtRjhGJ0Yn

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

KiZJInhHNiIiIiJJInlHRiRGJQ==

KiYpSSJ4RzYiIiIjIiIiKUkieUdGJUYmRic=

KiYpSSJ4RzYiIiIkIiIiKUkieUdGJUYmRic=

LCYqJiMiIiIiIiNGJSlJInhHNiIiIiRGJUYlKiZGJEYlKUkieUdGKUYqRiVGJQ==

LCYqJiMiIiIiIiNGJSlJInhHNiIiIidGJUYlKiZGJEYlKUkieUdGKUYqRiVGJQ==

LCYqKCMiIiIiIiNGJSlJInhHNiIiIiVGJUkieUdGKUYlRiUqKEYkRiVGKEYlKUYrRipGJUYl

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

KiYpSSJ4RzYiIiIjIiIiKUkieUdGJUYmRic=

KiYpSSJ4RzYiIiIlIiIiKUkieUdGJUYmRic=

LCYqJiMiIiIiIiNGJSlJInhHNiJGJkYlRiUqJkYkRiUpSSJ5R0YpRiZGJUYl

LCYqJiMiIiIiIiNGJSlJInhHNiIiIiVGJUYlKiZGJEYlKUkieUdGKUYqRiVGJQ==

LCYqJiMiIiIiIiNGJSlJInhHNiIiIidGJUYlKiZGJEYlKUkieUdGKUYqRiVGJQ==

LCYqJiMiIiIiIiNGJSlJInhHNiIiIilGJUYlKiZGJEYlKUkieUdGKUYqRiVGJQ==

LCYqKCMiIiIiIiNGJSlJInhHNiIiIiVGJSlJInlHRilGJkYlRiUqKEYkRiUpRihGJkYlKUYsRipGJUYl

LCYqKCMiIiIiIiNGJSlJInhHNiIiIidGJSlJInlHRilGJkYlRiUqKEYkRiUpRihGJkYlKUYsRipGJUYl

JSFH 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

KiYpSSJ4RzYiIiIjIiIiKUkieUdGJUYmRic=

KiYpSSJ4RzYiIiIlIiIiKUkieUdGJUYmRic=

LCYqJiMiIiIiIiNGJSlJInhHNiIiIiVGJUYlKiZGJEYlKUkieUdGKUYqRiVGJQ==

LCYqJiMiIiIiIiNGJSlJInhHNiIiIilGJUYlKiZGJEYlKUkieUdGKUYqRiVGJQ==

LCYqKCMiIiIiIiNGJUkieEc2IkYlKUkieUdGKCIiJkYlRiUqKEYkRiUpRidGK0YlRipGJSEiIg==

LCYqKCMiIiIiIiNGJSlJInhHNiIiIiZGJUkieUdGKUYlRiUqKEYkRiVGKEYlKUYrRipGJSEiIg==

LCYqKCMiIiIiIiNGJSlJInhHNiIiIidGJSlJInlHRilGJkYlRiUqKEYkRiUpRihGJkYlKUYsRipGJUYl

Molien's seriese indicate the number of linearly independent generators at each degree. By eliminating the algebraicaly dependent generators from the list of invariants given by the procedure invs we get the following invariant rings. 1) 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 2) 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 3) 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 4)) 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 11) 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 12) 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 14) 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 15) 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 To check if a given polynomial is invariant and to express an invariant polynomial in terms of the generators Method 1 When Reynolds (f) = f LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH invpol1:=proc(G,f); if evalb(Reynolds(G,f)=f) then print(invariant); else print(not invariant); fi; end proc;

Zio2JEkiR0c2IkkiZkdGJUYlRiVGJUAlLUkmZXZhbGJHJSpwcm90ZWN0ZWRHNiMvLUkpUmV5bm9sZHNHRiVGI0YmLUkmcHJpbnRHRio2I0kqaW52YXJpYW50R0YlLUYwNiM0RjJGJUYlRiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZIRitGKy1JI21pR0YkNidRKUV4YW1wbGVzRidGLy8lJ2l0YWxpY0dGOUYyRjVGLy8lK2V4ZWN1dGFibGVHRjEvJTBmb250X3N0eWxlX25hbWVHUSkyRH5JbnB1dEYnRjJGNQ== 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

SSppbnZhcmlhbnRHNiI=

JSFH 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

LCoqJilJInhHNiIiIiciIiIpSSJ5R0YmIiIkRihGKCooRitGKClGJSIiJkYoKUYqIiIlRighIiIqKEYrRigpRiVGMEYoKUYqRi5GKEYoKiYpRiVGK0YoKUYqRidGKEYx

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

SSppbnZhcmlhbnRHNiI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbW9HRiQ2L1EifkYnLyUlYm9sZEdRJXRydWVGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGNC8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGOS8lKXN0cmV0Y2h5R0Y5LyUqc3ltbWV0cmljR0Y5LyUobGFyZ2VvcEdGOS8lLm1vdmFibGVsaW1pdHNHRjkvJSdhY2NlbnRHRjkvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZILUkjbWlHRiQ2J1EnbWV0aG9kRidGLy8lJ2l0YWxpY0dGOUYyRjVGKy1JI21uR0YkNiZRIjJGJ0YvRjJGNUYvRjJGNQ== 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 G=reference for group V=list of variables [x, y] T=list of generators N=list of names to represent generators [\316\261, \316\262, \316\263,...] f is a polynomial If the polynomial obtained consists only \316\261, \316\262, \316\263,... then f is invariant , otherwise it is not invariant. The polynomial obtained is the expression of f in terms of the generators \316\261, \316\262, \316\263,... invpol2:=proc(G,V,T,N,f) local M,B,VV; M:=[seq(T[r]-N[r],r=1..nops(T))];VV:=[V[],N[]]; B:=Basis(M,tdeg(VV[])); NormalForm(f,B,tdeg(VV[])); end proc;

Zio2J0kiR0c2IkkiVkdGJUkiVEdGJUkiTkdGJUkiZkdGJTYlSSJNR0YlSSJCR0YlSSNWVkdGJUYlRiVDJj5GKzcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkLCYmRic2I0kickdGJSIiIiZGKEY3ISIiL0Y4O0Y5LUklbm9wc0dGMzYjRic+Ri03JCZGJkYlJkYoRiU+RiwtX0kpR3JvZWJuZXJHNiRGM0koX3N5c2xpYkdGJUkmQmFzaXNHRiU2JEYrLUkldGRlZ0dGJTYjJkYtRiUtX0ZISStOb3JtYWxGb3JtR0YlNiVGKUYsRk1GJUYlRiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHRic2J1EpRXhhbXBsZXNGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y0 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

LCwqJClJJmdhbW1hRyUqcHJvdGVjdGVkRyIiIyIiIkYoKigiIiVGKEklYmV0YUc2IkYoRiVGKEYoKigiIidGKEkmYWxwaGFHRixGKEYlRihGKCooRipGKEYvRihGK0YoRigqJClGL0YnRihGKA==

Relations G= reference for the group V=list of variables [x, y] T=list of generators N=list of names to represent generators [\316\261, \316\262, \316\263,...] The relations are given by the polynomials in \316\261, \316\262, \316\263,..obtained by the following procedure. JSFH relations:=proc(G,V,T,N) local M,B,VV,f; M:=[seq(T[r]-N[r],r=1..nops(T))];VV:=[V[],N[]]; B:=Basis(M,tdeg(VV[])); f:=q->is(indets(q) subset (convert(N,set)));select(f,B); end proc;

Zio2JkkiR0c2IkkiVkdGJUkiVEdGJUkiTkdGJTYmSSJNR0YlSSJCR0YlSSNWVkdGJUkiZkdGJUYlRiVDJz5GKjcjLUkkc2VxRyUqcHJvdGVjdGVkRzYkLCYmRic2I0kickdGJSIiIiZGKEY3ISIiL0Y4O0Y5LUklbm9wc0dGMzYjRic+Riw3JCZGJkYlJkYoRiU+RistX0kpR3JvZWJuZXJHNiRGM0koX3N5c2xpYkdGJUkmQmFzaXNHRiU2JEYqLUkldGRlZ0dGJTYjJkYsRiU+Ri1mKjYjSSJxR0YlRiU2JEkpb3BlcmF0b3JHRiVJJmFycm93R0YlRiUtSSNpc0dGSTYjLUknc3Vic2V0R0YzNiQtSSdpbmRldHNHRjNGUy1JKGNvbnZlcnRHRjM2JEYoSSRzZXRHRjNGJUYlNiRGKDknLUknc2VsZWN0R0YzNiRGLUYrRiVGJUYl

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NyI=

JSFH 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

NyI=

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

NyMsKComIiIlIiIiKUklYmV0YUc2IiIiIyIiIiEiIiomSSViZXRhRzYiIiIiKUkmYWxwaGFHNiIiIiMiIiIiIiIqJClJJmdhbW1hRyUqcHJvdGVjdGVkRyIiIyIiIiEiIg==

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

NyI=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= 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

NyI=

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

NyMsKComIiIlIiIiKUklYmV0YUc2IiIiJCIiIiEiIiomSSViZXRhRzYiIiIiKUkmYWxwaGFHNiIiIiMiIiIiIiIqJClJJmdhbW1hRyUqcHJvdGVjdGVkRyIiIyIiIiEiIg==

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYkLUkjbW9HRiQ2LVEifkYnLyUsbWF0aHZhcmlhbnRHUSdub3JtYWxGJy8lJmZlbmNlR1EmZmFsc2VGJy8lKnNlcGFyYXRvckdGNC8lKXN0cmV0Y2h5R0Y0LyUqc3ltbWV0cmljR0Y0LyUobGFyZ2VvcEdGNC8lLm1vdmFibGVsaW1pdHNHRjQvJSdhY2NlbnRHRjQvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZDRi8= Thus we have the following invariant rings 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The other invariant rings can be found in the same way. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Remark These procedures can be extended to inclue the case of three variables x, y, z. One may also try to automate the finding of the algebraically independent generators. e-mail: kahtanalzubaidy@yahoo.com Department of Mathematics, Faculty of Science, University of Benghazi, Libya. 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NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIyEiIiIiIywkKiQiIiQjIiIiRilGLUYqRi1GJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTkwMDk0NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIyEiIiIiIywkKiQiIiQjIiIiRilGJ0YqRi1GJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTkwMTA0NFgsJSlhbnl0aGluZ0c2IyUpaWRlbnRpdHlHNiJbZ2whIiIhISEjISIjIiM2Ig==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTkwMTI0NFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjJSJ3RyIiIUYoKiRGJyIiI0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzOTkwMTMyNFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjKiQlIndHIiIjIiIhRioqJEYoIiIlRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzOTkzNDMyNFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhIiIiRihGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzOTkzNDQwNFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhKiQlIndHIiIjRilGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzOTkzNDUwNFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhKiQlIndHIiIlKiRGKSIiI0YnRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzOTkwMTA0NFgsJSlhbnl0aGluZ0c2IyUpaWRlbnRpdHlHNiJbZ2whIiIhISEjISIjIiM2Ig==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TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk1MTU5NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhIiIiISIiRidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk1MTY3NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjISIiIiIhRihGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzOTk1MTc3NlgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhISIiIiIiRidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk1MjA0OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIiIiIhRighIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk1MjMwMFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhIiIiRihGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzOTk1MjM4MFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjISIiIiIhRigiIiJGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk1MjU4MFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhISIiRihGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzOTk1MjY2MFgsJSlhbnl0aGluZ0c2IyUpaWRlbnRpdHlHNiJbZ2whIiIhISEjISIjIiM2Ig==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk2OTI2OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhISIiIiIiRidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk2OTQ0OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjISIiIiIhRihGJ0YlTTdSMApJNlJUQUJMRV9TQVZFLzEzOTk2OTUyOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhIiIiISIiRidGJQ==TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk2OTYyOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhXiMiIiJGKEYnRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk4NjEyOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjXiMiIiIiIiFGKV4jISIiRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk4NjMwOFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjIiIhXiMhIiJGKEYnRiU=TTdSMApJNlJUQUJMRV9TQVZFLzEzOTk4NjQ4OFgsJSlhbnl0aGluZ0c2IjYiW2dsISIlISEhIyUiIyIjXiMhIiIiIiFGKV4jIiIiRiU=