JSFH JSFH Quotient Polynomial Rings by Maple Kahtan H. Alzubaidy Department of Mathematics, Faculty of Science, University of Benghazi E-mail: kahtanalzubaidy@yahoo.com Key Words: Quotient Rings, Multivariate Polynomials, Groebner Basis Introduction. Quotient polynomial rings over the infinite field containing \342\204\232 are involved. The computations concern the univariate polynomial rings and the multivariate polynomial rings in two variables. In both cases a vector space basis for the quotient is constructed. The case of of several variables includes cases of infinite dimensions. Procedure II-2 recognizes the infinite dimensional cases , however we shall restrict ourselves to the finite computations. Ring operations of addition and multiplication on the quotients are computed as well. Goebner basis is used and the computations are carried out in Maple 13. I) Univariate Polynomials. Let F be a field. The polynomialring F[x] is a principal ideal domain. If I is an ideal of F[x], then I=<f(x)> for some f(x)\342\210\210F[x]. We have the quotient polynomial ring F[x]/I. F[x]/I={r(x): r(x) is the remainder of any polynomial in F[x] when it is divided by f(x)}. Theorem 1. If the degree of f(x) is n, then F[x]/I is an n-dimensional vector space over F with basis {1,x,..,LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2KFEieEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR1EmZmFsc2VGJy8lLG1hdGh2YXJpYW50R1ElYm9sZEYnLyUrZm9udHdlaWdodEdGPS1GIzYrLUYvNihRIm5GJ0YyRjVGOEY7Rj4tSSNtb0dGJDYwUSgmbWludXM7RidGMkY1RjtGPi8lJmZlbmNlR0Y6LyUqc2VwYXJhdG9yR0Y6LyUpc3RyZXRjaHlHRjovJSpzeW1tZXRyaWNHRjovJShsYXJnZW9wR0Y6LyUubW92YWJsZWxpbWl0c0dGOi8lJ2FjY2VudEdGOi8lJ2xzcGFjZUdRLDAuMjIyMjIyMmVtRicvJSdyc3BhY2VHRlktSSNtbkdGJDYnUSIxRidGMkY1RjtGPkYyRjUvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y3RjtGPi8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMkY1RjtGPg==}. The ring operations of F[x]/I are given as follows 1) addition 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 F[x]/I 2) multiplication 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 Theorem 2. F[x]/I is a field iff f(x) is irreducible over F. The inverse of r(x) in the field F[x]/I is given as follows gcd(f(x),r(x))=u(x)f(x)+v(x)r(x)=1 for some u(x), v(x)\342\210\210F[x]. Therefore v(x)r(x)=1 in F[x]/I. Thus inverse(r(x))=v(x). Proceduers Example LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUkjbWlHRiQ2JlEocmVzdGFydEYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEiO0YnRi8vRjZRJ25vcm1hbEYnLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkAvJSpzeW1tZXRyaWNHRkAvJShsYXJnZW9wR0ZALyUubW92YWJsZWxpbWl0c0dGQC8lJ2FjY2VudEdGQC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRi8vJStleGVjdXRhYmxlR0ZARjw= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYtLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGLy9GNlEnbm9ybWFsRicvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkAvJSlzdHJldGNoeUdGQC8lKnN5bW1ldHJpY0dGQC8lKGxhcmdlb3BHRkAvJS5tb3ZhYmxlbGltaXRzR0ZALyUnYWNjZW50R0ZALyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGTy1JJW1zdXBHRiQ2JS1GLDYmUSJ4RidGL0YyRjUtRiM2Ji1JI21uR0YkNiVRIjNGJ0YvRjxGL0YyRjUvJTFzdXBlcnNjcmlwdHNoaWZ0R1EiMEYnLUY5Ni5RIitGJ0YvRjxGPkZBRkNGRUZHRklGSy9GTlEsMC4yMjIyMjIyZW1GJy9GUUZfb0ZVRltvLUZlbjYlUSIxRidGL0Y8LUY5Ni5RIjtGJ0YvRjxGPi9GQkY0RkNGRUZHRklGSy9GTlEmMC4wZW1GJ0ZQRi8vJStleGVjdXRhYmxlR0ZARjw=

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SSV0cnVlRyUqcHJvdGVjdGVkRw==

JSFH inv:=proc(A,f); gcdex(A,f,x,'u','v'); sort(u,x); end proc;

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 Theorem 3. 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 Corollory 4. 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, and 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.Thus we have Theorem 5. 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. Corollory 6. 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 Proceduers Procedure II.1 Leading Monomials LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYpLUkjbWlHRiQ2KFEocmVzdGFydEYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y0LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1JI21vR0YkNjBRIjtGJ0YvRjIvRjhGPEY6LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y0LyUpc3RyZXRjaHlHRkQvJSpzeW1tZXRyaWNHRkQvJShsYXJnZW9wR0ZELyUubW92YWJsZWxpbWl0c0dGRC8lJ2FjY2VudEdGRC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHUSwwLjI3Nzc3NzhlbUYnRi9GMi8lK2V4ZWN1dGFibGVHRkRGQUY6 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= lm:=proc() local L,G,h,hh,H; L:= [seq(args[i],i=1..nargs)]; G:=Basis(L,tdeg(x,y)); h:=LeadingMonomial(G,tdeg(x,y));if nops(h)=1 then h; else hh:=sum(h[j],j=1..nops(h)); H:=sort(hh,order=plex(x,y)); [seq(op(s,H),s=1..nops(H))];fi end proc;

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JSFH JSFH Procedure II.2 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 fin:=proc() local K; K:=lm(seq(args[i],i=1..nargs)); if gcd(K[1],K[nops(K)])=1 then print("quotient is finite dimentional")else print("quotient is infinite dimensional") fi; end proc;

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JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYqLUkjbW9HRiQ2MFEifkYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y3LyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0Y8LyUpc3RyZXRjaHlHRjwvJSpzeW1tZXRyaWNHRjwvJShsYXJnZW9wR0Y8LyUubW92YWJsZWxpbWl0c0dGPC8lJ2FjY2VudEdGPC8lJ2xzcGFjZUdRJjAuMGVtRicvJSdyc3BhY2VHRktGKy1JI21pR0YkNihRKUV4YW1wbGVzRidGL0YyLyUnaXRhbGljR0Y8RjVGOEYyLyUrZXhlY3V0YWJsZUdGNC8lMGZvbnRfc3R5bGVfbmFtZUdRKTJEfklucHV0RidGNUY4 JSFH JSFH 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

LCYqJkkieEc2IiIiIilJInlHRiUiIiRGJkYmKiQpRiQiIiNGJiEiIg==

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UT9xdW90aWVudH5pc35maW5pdGV+ZGltZW50aW9uYWw2Ig==

JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiUtSSNtbkdGJDYlUSIxRidGMi9GOVEnbm9ybWFsRidGMkZBLyUvc3Vic2NyaXB0c2hpZnRHUSIwRictSSNtb0dGJDYuUSomY29sb25lcTtGJ0YyRkEvJSZmZW5jZUdRJmZhbHNlRicvJSpzZXBhcmF0b3JHRkwvJSlzdHJldGNoeUdGTC8lKnN5bW1ldHJpY0dGTC8lKGxhcmdlb3BHRkwvJS5tb3ZhYmxlbGltaXRzR0ZMLyUnYWNjZW50R0ZMLyUnbHNwYWNlR1EsMC4yNzc3Nzc4ZW1GJy8lJ3JzcGFjZUdGZW4tSSVtc3VwR0YkNiUtRi82JlEieEYnRjJGNUY4LUYjNiUtRj42JVEiM0YnRjJGQUYyRkEvJTFzdXBlcnNjcmlwdHNoaWZ0R0ZFLUZHNi5RIitGJ0YyRkFGSkZNRk9GUUZTRlVGVy9GWlEsMC4yMjIyMjIyZW1GJy9GZ25GaW8tRmluNiVGW28tRiM2JS1GPjYlUSIyRidGMkZBRjJGQUZjby1GRzYuUScmc2RvdDtGJ0YyRkFGSkZNRk9GUUZTRlVGVy9GWlEmMC4wZW1GJy9GZ25GZnAtRi82JlEieUYnRjJGNUY4RmVvLUZpbjYlRmhwRl5vRmNvLUZHNi5RIjtGJ0YyRkFGSi9GTkY3Rk9GUUZTRlVGV0ZlcEZmbi1GRzYuUSJ+RidGMkZBRkpGTUZPRlFGU0ZVRldGZXBGZ3AtRiw2JUYuRl1wRkNGRi1GaW42JS1GLzYmRl1vRjIvRjZGTEZBRl1wRmNvRmVvLUZpbjYlLUYvNiZGanBGMkZqcUZBRl1wRmNvLUZHNi1GX3FGQUZKRmBxRk9GUUZTRlVGV0ZlcEZmbi8lK2V4ZWN1dGFibGVHRkxGQQ==

LCgqJClJInhHNiIiIiQiIiJGKComKUYlIiIjRihJInlHRiZGKEYoKiQpRixGJ0YoRig=

LCYqJClJInhHNiIiIiMiIiJGKCokKUkieUdGJkYnRihGKA==

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

UT9xdW90aWVudH5pc35maW5pdGV+ZGltZW50aW9uYWw2Ig==

JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYoLUkjbWlHRiQ2JlEkZmluRicvJSVzaXplR1EjMTRGJy8lJ2l0YWxpY0dRJXRydWVGJy8lLG1hdGh2YXJpYW50R1EnaXRhbGljRictSShtZmVuY2VkR0YkNiUtRiM2Jy1JJW1zdXBHRiQ2JS1GLDYmUSJ4RidGLy9GM1EmZmFsc2VGJy9GNlEnbm9ybWFsRictRiM2JS1JI21uR0YkNiVRIjNGJ0YvRkVGL0ZFLyUxc3VwZXJzY3JpcHRzaGlmdEdRIjBGJy1JI21vR0YkNi5RIitGJ0YvRkUvJSZmZW5jZUdGRC8lKnNlcGFyYXRvckdGRC8lKXN0cmV0Y2h5R0ZELyUqc3ltbWV0cmljR0ZELyUobGFyZ2VvcEdGRC8lLm1vdmFibGVsaW1pdHNHRkQvJSdhY2NlbnRHRkQvJSdsc3BhY2VHUSwwLjIyMjIyMjJlbUYnLyUncnNwYWNlR0Zeby1GPjYlLUYsNiZRInlGJ0YvRkNGRUZHRk1GL0ZFRi9GRS1GUTYuUSI7RidGL0ZFRlQvRldGNEZYRlpGZm5GaG5Gam4vRl1vUSYwLjBlbUYnL0Zgb1EsMC4yNzc3Nzc4ZW1GJ0YvLyUrZXhlY3V0YWJsZUdGREZF

UUFxdW90aWVudH5pc35pbmZpbml0ZX5kaW1lbnNpb25hbDYi

JSFH Procedure II.3 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 JSFH tm:=proc() local K,p,q; K:=lm(seq(args[i],i=1..nargs)); q:=expand(sum((op(1,K[1]))^i,i=0..op(2,K[1])-1)* sum((op(1,K[nops(K)]))^j,j=0..op(2,K[nops(K)])-1)); {seq(op(r, q), r = 1 .. nops(q))}; end proc;

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JSFH 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 JSFH 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

LCYqJkkieEc2IiIiIilJInlHRiUiIiRGJkYmKiQpRiQiIiNGJiEiIg==

LCYqJilJInhHNiIiIiQiIiIpSSJ5R0YmIiIjRihGKEYqISIi

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

PDIiIiJJInhHNiJJInlHRiUqJClGJCIiI0YjKiQpRiQiIiRGIyokKUYmRilGIyokKUYmRixGIyomRiRGI0YmRiMqJkYkRiNGLkYjKiZGJEYjRjBGIyomRihGI0YmRiMqJkYoRiNGLkYjKiZGKEYjRjBGIyomRitGI0YmRiMqJkYrRiNGLkYjKiZGK0YjRjBGIw==

IiM7

JSFH 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

LCgqJClJInhHNiIiIiQiIiJGKComKUYlIiIjRihJInlHRiZGKEYoKiQpRixGJ0YoRig=

LCYqJClJInhHNiIiIiMiIiJGKCokKUkieUdGJkYnRihGKA==

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

PCoiIiJJInhHNiJJInlHRiUqJClGJiIiI0YjKiQpRiYiIiRGIyomRiRGI0YmRiMqJkYkRiNGKEYjKiZGJEYjRitGIw==

IiIp

JSFH Procedure II.4 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 rm:=proc(w,A) local bn; bn:=u->is(divide(u,w)); remove(bn,A); end proc;

Zio2JEkid0c2IkkiQUdGJTYjSSNibkdGJUYlRiVDJD5GKGYqNiNJInVHRiVGJTYkSSlvcGVyYXRvckdGJUkmYXJyb3dHRiVGJS1JI2lzR0YlNiMtSSdkaXZpZGVHJSpwcm90ZWN0ZWRHNiRGLUYkRiVGJTYkRiQ5JC1JJ3JlbW92ZUdGNjYkRihGJkYlRiVGJQ==

JSFH Procedure II.5 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYrLUkjbWlHRiQ2KFEnVmVjdG9yRicvJSVzaXplR1EjMTRGJy8lJWJvbGRHUSV0cnVlRicvJSdpdGFsaWNHUSZmYWxzZUYnLyUsbWF0aHZhcmlhbnRHUSVib2xkRicvJStmb250d2VpZ2h0R0Y6LUkjbW9HRiQ2MFEifkYnRi9GMkY4RjsvJSZmZW5jZUdGNy8lKnNlcGFyYXRvckdGNy8lKXN0cmV0Y2h5R0Y3LyUqc3ltbWV0cmljR0Y3LyUobGFyZ2VvcEdGNy8lLm1vdmFibGVsaW1pdHNHRjcvJSdhY2NlbnRHRjcvJSdsc3BhY2VHUSYwLjBlbUYnLyUncnNwYWNlR0ZRLUYsNihRJnNwYWNlRidGL0YyRjVGOEY7Rj0tRiw2KFEmYmFzaXNGJ0YvRjJGNUY4RjtGL0YyRjhGOw== JSFH vsb:=proc() local LM,B,A,w; LM:=lm(seq(args[i],i=1..nargs)); B:={seq(LM[t],t=2..nops(LM)-1)};A:=tm(seq(args[i],i=1..nargs)); for w in B do A:=rm(w,A) od: A; end proc;

Zio2IjYmSSNMTUdGI0kiQkdGI0kiQUdGI0kid0dGI0YjRiNDJz5GJS1JI2xtR0YjNiMtSSRzZXFHJSpwcm90ZWN0ZWRHNiQmJSVhcmdzRzYjSSJpR0YjL0Y1OyIiIiUmbmFyZ3NHPkYmPCMtRi82JCZGJTYjSSJ0R0YjL0ZAOyIiIywmLUklbm9wc0dGMDYjRiVGOEY4ISIiPkYnLUkjdG1HRiNGLT8mRihGJkkldHJ1ZUdGMD5GJy1JI3JtR0YjNiRGKEYnRidGI0YjRiM=

JSFH 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 JSFH 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

LCYqJkkieEc2IiIiIilJInlHRiUiIiRGJkYmKiQpRiQiIiNGJiEiIg==

LCYqJilJInhHNiIiIiQiIiIpSSJ5R0YmIiIjRihGKEYqISIi

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

PC4iIiJJInhHNiJJInlHRiUqJClGJCIiI0YjKiQpRiQiIiRGIyokKUYmRilGIyokKUYmRixGIyomRiRGI0YmRiMqJkYkRiNGLkYjKiZGKEYjRiZGIyomRihGI0YuRiMqJkYrRiNGJkYj

IiM3

JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzY1LUklbXN1YkdGJDYlLUkjbWlHRiQ2JlEiZkYnLyUlc2l6ZUdRIzE0RicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0YnLUYjNiQtSSNtbkdGJDYlUSIxRidGMi9GOVEnbm9ybWFsRidGQS8lL3N1YnNjcmlwdHNoaWZ0R1EiMEYnLUkjbW9HRiQ2LlEqJmNvbG9uZXE7RidGMkZBLyUmZmVuY2VHUSZmYWxzZUYnLyUqc2VwYXJhdG9yR0ZMLyUpc3RyZXRjaHlHRkwvJSpzeW1tZXRyaWNHRkwvJShsYXJnZW9wR0ZMLyUubW92YWJsZWxpbWl0c0dGTC8lJ2FjY2VudEdGTC8lJ2xzcGFjZUdRLDAuMjc3Nzc3OGVtRicvJSdyc3BhY2VHRmVuLUklbXN1cEdGJDYlLUYvNiZRInhGJ0YyRjVGOC1GIzYkLUY+NiVRIjNGJ0YyRkFGQS8lMXN1cGVyc2NyaXB0c2hpZnRHRkUtRkc2LlEiK0YnRjJGQUZKRk1GT0ZRRlNGVUZXL0ZaUSwwLjIyMjIyMjJlbUYnL0ZnbkZpby1GaW42JUZbby1GIzYkLUY+NiVRIjJGJ0YyRkFGQUZjby1GRzYuUScmc2RvdDtGJ0YyRkFGSkZNRk9GUUZTRlVGVy9GWlEmMC4wZW1GJy9GZ25GZnAtRi82JlEieUYnRjJGNUY4RmVvLUZpbjYlRmhwRl5vRmNvLUZHNi5RIjtGJ0YyRkFGSi9GTkY3Rk9GUUZTRlVGV0ZlcEZmbi1GRzYuUSJ+RidGMkZBRkpGTUZPRlFGU0ZVRldGZXBGZ3AtRiw2JUYuRl1wRkNGRi1GaW42JS1GLzYmRl1vRjIvRjZGTEZBRl1wRmNvRmVvLUZpbjYlLUYvNiZGanBGMkZqcUZBRl1wRmNvLUZHNi1GX3FGQUZKRmBxRk9GUUZTRlVGV0ZlcEZmbi8lK2V4ZWN1dGFibGVHRkxGQQ==

LCgqJClJInhHNiIiIiQiIiJGKComKUYlIiIjRihJInlHRiZGKEYoKiQpRixGJ0YoRig=

LCYqJClJInhHNiIiIiMiIiJGKCokKUkieUdGJkYnRihGKA==

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

PCgiIiJJInhHNiJJInlHRiUqJClGJiIiI0YjKiQpRiYiIiRGIyomRiRGI0YmRiM=

IiIn

JSFH LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Ring Operations Addition Procedure II.6 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=adm:=proc(A,B,L) local G; G:=Basis(L,tdeg(x,y)); NormalForm(A+B,G,tdeg(x,y)); end proc;

Zio2JUkiQUc2IkkiQkdGJUkiTEdGJTYjSSJHR0YlRiVGJUMkPkYpLV9JKUdyb2VibmVyRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlSSZCYXNpc0dGJTYkRictSSV0ZGVnR0YlNiRJInhHRiVJInlHRiUtX0YuSStOb3JtYWxGb3JtR0YlNiUsJkYkIiIiRiZGPkYpRjRGJUYlRiU=

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= JSFH 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

LCgqJClJInhHNiIiIiQiIiJGKComKUYlIiIjRihJInlHRiZGKEYoKiQpRixGJ0YoRig=

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

LC4iIiMiIiJJInhHNiIhIiIqJiIiJEYkSSJ5R0YmRiRGJComIiImRiQpRipGI0YkRiQqJiIiJUYkKUYqRilGJEYnKigiIidGJEYlRiRGKkYkRiQ=

addition A+B 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

LCwiIiQiIiJJInhHNiJGJComIiIpRiRJInlHRiZGJEYkKiYiIiNGJClGKUYrRiRGJCooIiIlRiRGJUYkRilGJEYk

multiplication AB 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

LC4iIiMiIiIqJiIjOEYkSSJ5RzYiRiRGJComIiIkRiRJInhHRihGJEYkKiYiIztGJClGJ0YjRiRGJCooRipGJEYrRiRGJ0YkRiQqJiIiJ0YkKUYnRipGJEYk

square LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYnLUklbXN1cEdGJDYlLUkjbWlHRiQ2KFEiQUYnLyUlc2l6ZUdRIzE0RicvJSVib2xkR1EldHJ1ZUYnLyUnaXRhbGljR0Y3LyUsbWF0aHZhcmlhbnRHUSxib2xkLWl0YWxpY0YnLyUrZm9udHdlaWdodEdRJWJvbGRGJy1GIzYqLUkjbW5HRiQ2J1EiMkYnRjJGNS9GO0Y/Rj1GMkY1RjgvJStmb3JlZ3JvdW5kR1EsWzIwMCwwLDIwMF1GJy8lLHBsYWNlaG9sZGVyR0Y3RjpGPS8lMXN1cGVyc2NyaXB0c2hpZnRHUSIwRidGMkY1RkZGPQ== 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

LC4iIiJGIyomIiM1RiNJInlHNiJGI0YjKiYiIiVGI0kieEdGJ0YjRiMqJiIjOkYjKUYmIiIjRiNGIyooIiM7RiNGKkYjRiZGI0YjKiYiIzlGIylGJiIiJEYjISIi

LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic= Remark The above procedures can easily be generalized to include the case of three variables x, y, z. LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2I1EhRic=