Komputeralgebrai algoritmusok J\303\241rai Antal Ezek a programok csak szeml\303\251ltet\303\251sre szolg\303\241lnak.
<Text-field style="Heading 1" layout="Heading 1">1. T<Font encoding="UTF-8">\303\266rt\303\251</Font>net</Text-field>
<Text-field style="Heading 1" layout="Heading 1">2. Algebrai alapok</Text-field>
<Text-field style="Heading 1" layout="Heading 1">3. Norm<Font encoding="UTF-8">\303\241</Font>l form<Font encoding="UTF-8">\303\241</Font>k, reprezent<Font encoding="UTF-8">\303\241ci\303\263</Font></Text-field>
<Text-field style="Heading 1" layout="Heading 1">4. Aritmetika</Text-field>
<Text-field style="Heading 1" layout="Heading 1">5. K<Font encoding="UTF-8">\303\255</Font>nai marad<Font encoding="UTF-8">\303\251</Font>kol<Font encoding="UTF-8">\303\241</Font>s</Text-field>
<Text-field style="Heading 1" layout="Heading 1">6. Newton-iter<Font encoding="UTF-8">\303\241</Font>ci<Font encoding="UTF-8">\303\263</Font>, Hensel-felemel<Font encoding="UTF-8">\303\251</Font>s</Text-field> restart;
<Text-field style="Heading 2" layout="Heading 2">E 6.1. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> `mod`:=mods; p:=97; u:=-272300; u0:=u mod p; u1:=(u-u0)/p mod p; u2:=(u-(u0+u1*p))/p^2 mod p; u-(u0+u1*p+u2*p^2); SSVtb2RzRyUqcHJvdGVjdGVkRw== IiMoKg== IScrQkY= ISNA IiIn ISNI IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 6.2. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=5; u:=14*x^2-11*x-15; u0:=u mod p; u1:=(u-u0)/p mod p; u2:=(u-(u0+u1*p))/p^2 mod p; u-(u0+u1*p+u2*p^2); IiIm LCgqJiIjOSIiIilJInhHNiIiIiNGJUYlKiYiIzZGJUYnRiUhIiIiIzpGLA== LCYqJClJInhHNiIiIiMiIiIhIiJGJUYp LCgqJiIiIyIiIilJInhHNiJGJEYlISIiKiZGJEYlRidGJUYpRiRGJQ== LCYqJClJInhHNiIiIiMiIiJGKEYoISIi IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 6.3. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> u:='u'; a:=36*x^4-180*x^3+93*x^2+330*x+121; F:=a-u^2; Fp:=diff(F,u); SSJ1RzYi LCwqJiIjTyIiIilJInhHNiIiIiVGJUYlKiYiJCE9RiUpRiciIiRGJSEiIiomIiMkKkYlKUYnIiIjRiVGJSomIiRJJEYlRidGJUYlIiRAIkYl LC4qJiIjTyIiIilJInhHNiIiIiVGJUYlKiYiJCE9RiUpRiciIiRGJSEiIiomIiMkKkYlKUYnIiIjRiVGJSomIiRJJEYlRidGJUYlIiRAIkYlKiQpSSJ1R0YoRjJGJUYu LCQqJiIiIyIiIkkidUc2IkYlISIi a mod p; LCgqJClJInhHNiIiIiUiIiJGKComIiIjRigpRiVGKkYoISIiRihGKA== u0:=x^2-1; u[1]:=u0; d:=subs(u=u[1],Fp); LCYqJClJInhHNiIiIiMiIiJGKEYoISIi LCYqJClJInhHNiIiIiMiIiJGKEYoISIi LCYqJiIiIyIiIilJInhHNiJGJEYlISIiRiRGJQ== u1:=-expand(subs(u=u[1],F)/p); u1:=Quo(u1,d,x) mod p; LCwqJiIiKCIiIilJInhHNiIiIiVGJSEiIiomIiNPRiUpRiciIiRGJUYlKiYiIz5GJSlGJyIiI0YlRioqJiIjbUYlRidGJUYqIiNDRio= LCgqJClJInhHNiIiIiMiIiJGKComRidGKEYlRihGKEYnISIi u[2]:=u0+5*u1; u2:=-expand(subs(u=u[2],F)/p^2); u2:=Quo(u2,d,x) mod p; LCgqJiIiJyIiIilJInhHNiIiIiNGJUYlIiM2ISIiKiYiIzVGJUYnRiVGJQ== LCgqJiIjNyIiIilJInhHNiIiIiRGJUYlKiYiIiZGJSlGJyIiI0YlISIiKiYiI0FGJUYnRiVGLg== LCRJInhHNiIhIiI= u[3]:=u0+5*u1+5^2*u2; expand(subs(u=u[3],F)); LCgqJiIiJyIiIilJInhHNiIiIiNGJUYlIiM2ISIiKiYiIzpGJUYnRiVGKw== IiIh
<Text-field style="Heading 2" layout="Heading 2">E 6.4. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=5; `mod`:=mods; a:=x^4+x^3*y^2-x^2*y^4+x^2*y*z+2*x^2*z-2*x^2-2*x*y^3*z+x*y^2*z-x*y^2-y^2*z^2+y*z^2-y*z+z^2-2*z+1 mod p; a:=collect(a,[y,z],`distributed`); sort(a,[y,z],tdeg); F:=a-u^2; Fp:=diff(F,u); IiIm SSVtb2RzRyUqcHJvdGVjdGVkRw== LEAqJClJInhHNiIiIiUiIiJGKComKUYlIiIkRigpSSJ5R0YmIiIjRihGKComKUYlRi5GKClGLUYnRighIiIqKEYwRihGLUYoSSJ6R0YmRihGKCooRi5GKEYwRihGNEYoRigqJkYuRihGMEYoRjIqKkYuRihGJUYoKUYtRitGKEY0RihGMiooRiVGKEYsRihGNEYoRigqJkYlRihGLEYoRjIqJkYsRigpRjRGLkYoRjIqJkYtRihGPEYoRigqJkYtRihGNEYoRjIqJEY8RihGKComRi5GKEY0RihGMkYoRig= LDoqJClJInhHNiIiIiUiIiJGKComIiIjRigpRiVGKkYoISIiRihGKCoqRipGKEYlRigpSSJ5R0YmIiIkRihJInpHRiZGKEYsKihGJUYoKUYvRipGKEYxRihGKComRjNGKClGMUYqRihGLComRi9GKEY1RihGKCooLCYqJEYrRihGKEYoRixGKEYvRihGMUYoRigqJiwmRilGKEYqRixGKEYxRihGKComLCZGJUYsKiQpRiVGMEYoRihGKEYzRihGKCokRjVGKEYoKiZGK0YoKUYvRidGKEYs LDoqJilJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRighIiIqKkYnRihGJUYoKUYqIiIkRihJInpHRiZGKEYsKiYpRipGJ0YoKUYwRidGKEYsKihGJUYoRjJGKEYwRihGKComRipGKEYzRihGKComLCZGJUYsKiQpRiVGL0YoRihGKEYyRihGKCooLCYqJEYkRihGKEYoRixGKEYqRihGMEYoRigqJEYzRihGKComLCYqJkYnRihGJEYoRihGJ0YsRihGMEYoRigqJClGJUYrRihGKEZARixGKEYo LDwqJilJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRighIiIqKkYnRihGJUYoKUYqIiIkRihJInpHRiZGKEYsKiYpRipGJ0YoKUYwRidGKEYsKihGJUYoRjJGKEYwRihGKComRipGKEYzRihGKComLCZGJUYsKiQpRiVGL0YoRihGKEYyRihGKCooLCYqJEYkRihGKEYoRixGKEYqRihGMEYoRigqJEYzRihGKComLCYqJkYnRihGJEYoRihGJ0YsRihGMEYoRigqJClGJUYrRihGKEZARixGKEYoKiQpSSJ1R0YmRidGKEYs LCQqJiIiIyIiIkkidUc2IkYlISIi subs(y=0,z=0,a); u[1]:=x^2-1; LCgqJClJInhHNiIiIiUiIiJGKComIiIjRigpRiVGKkYoISIiRihGKA== LCYqJClJInhHNiIiIiMiIiJGKEYoISIi d:=subs(u=u[1],Fp) mod p; LCYqJiIiIyIiIilJInhHNiJGJEYlISIiRiRGJQ== FF:=expand(subs(u=u[1],F)) mod p; FF:=collect(%,[y,z],`distributed`); sort(%,[y,z],tdeg); u2:=0; u3:=-Quo(2*x^2-2,d,x) mod p; du[2]:=u2*y+u3*z; u[2]:=u[1]+du[2]; LDoqJilJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRighIiIqKkYnRihGJUYoKUYqIiIkRihJInpHRiZGKEYsKiYpRipGJ0YoKUYwRidGKEYsKihGJUYoRjJGKEYwRihGKComRipGKEYzRihGKComRiVGKEYyRihGLComKUYlRi9GKEYyRihGKCooRiRGKEYqRihGMEYoRigqJkYqRihGMEYoRiwqJEYzRihGKCooRidGKEYkRihGMEYoRigqJkYnRihGMEYoRiw= LDQqKiIiIyIiIkkieEc2IkYlKUkieUdGJyIiJEYlSSJ6R0YnRiUhIiIqKEYmRiUpRilGJEYlRitGJUYlKiZGLkYlKUYrRiRGJUYsKiZGKUYlRjBGJUYlKigsJiokKUYmRiRGJUYlRiVGLEYlRilGJUYrRiVGJSomLCYqJkYkRiVGNUYlRiVGJEYsRiVGK0YlRiUqJiwmRiZGLCokKUYmRipGJUYlRiVGLkYlRiUqJEYwRiVGJSomRjVGJSlGKSIiJUYlRiw= LDQqJilJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRighIiIqKkYnRihGJUYoKUYqIiIkRihJInpHRiZGKEYsKiYpRipGJ0YoKUYwRidGKEYsKihGJUYoRjJGKEYwRihGKComRipGKEYzRihGKComLCZGJUYsKiQpRiVGL0YoRihGKEYyRihGKCooLCYqJEYkRihGKEYoRixGKEYqRihGMEYoRigqJEYzRihGKComLCYqJkYnRihGJEYoRihGJ0YsRihGMEYoRig= IiIh IiIi SSJ6RzYi LCgqJClJInhHNiIiIiMiIiJGKEYoISIiSSJ6R0YmRig= FF:=expand(subs(u=u[2],F)) mod p; FF:=collect(%,[y,z],`distributed`); sort(%,[y,z],tdeg); u22:=-Quo(x^3-x,d,x) mod p; u23:=-Quo(x^2-1,d,x) mod p; u33:=-Quo(0,d,x) mod p; du[3]:=u22*y^2+u23*y*z+u33*z^2; u[3]:=u[2]+du[3]; LDQqJilJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRighIiIqKkYnRihGJUYoKUYqIiIkRihJInpHRiZGKEYsKiYpRipGJ0YoKUYwRidGKEYsKihGJUYoRjJGKEYwRihGKComRipGKEYzRihGKComRiVGKEYyRihGLComKUYlRi9GKEYyRihGKCooRiRGKEYqRihGMEYoRigqJkYqRihGMEYoRiw= LDAqKiIiIyIiIkkieEc2IkYlKUkieUdGJyIiJEYlSSJ6R0YnRiUhIiIqKEYmRiUpRilGJEYlRitGJUYlKiZGLkYlKUYrRiRGJUYsKiZGKUYlRjBGJUYlKigsJiokKUYmRiRGJUYlRiVGLEYlRilGJUYrRiVGJSomLCZGJkYsKiQpRiZGKkYlRiVGJUYuRiVGJSomRjVGJSlGKSIiJUYlRiw= LDAqJilJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRighIiIqKkYnRihGJUYoKUYqIiIkRihJInpHRiZGKEYsKiYpRipGJ0YoKUYwRidGKEYsKihGJUYoRjJGKEYwRihGKComRipGKEYzRihGKComLCZGJUYsKiQpRiVGL0YoRihGKEYyRihGKCooLCYqJEYkRihGKEYoRixGKEYqRihGMEYoRig= LCQqJiIiIyIiIkkieEc2IkYlISIi ISIj IiIh LCYqKCIiIyIiIkkieEc2IkYlKUkieUdGJ0YkRiUhIiIqKEYkRiVGKUYlSSJ6R0YnRiVGKg== LCwqJClJInhHNiIiIiMiIiJGKEYoISIiSSJ6R0YmRigqKEYnRihGJUYoKUkieUdGJkYnRihGKSooRidGKEYtRihGKkYoRik= FF:=expand(subs(u=u[3],F)) mod p; IiIh
<Text-field style="Heading 2" layout="Heading 2">E 6.5. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=5; m:=p; a:=x^3+10*x^2-432*x+5040; a mod p; u:=x; w:=x^2-2; e:=expand(a-u*w); IiIm IiIm LCoqJClJInhHNiIiIiQiIiJGKComIiM1RigpRiUiIiNGKEYoKiYiJEslRihGJUYoISIiIiVTXUYo LCYqJClJInhHNiIiIiQiIiJGKComIiIjRihGJUYoISIi SSJ4RzYi LCYqJClJInhHNiIiIiMiIiJGKEYnISIi LCgqJiIjNSIiIilJInhHNiIiIiNGJUYlKiYiJEklRiVGJ0YlISIiIiVTXUYl Gcdex(u,w,x,'s','t') mod p; s; t; IiIi LCQqJiIiIyIiIkkieEc2IkYlISIi IiIj c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCgqJiIiIyIiIilJInhHNiJGJEYlRiUqJiIjJylGJUYnRiUhIiIiJTM1RiU= LCgqJClJInhHNiIiIiQiIiJGKComIiIjRigpRiVGKkYoRihGJSEiIg== LCgqJClJInhHNiIiIiMiIiIhIiIqJkYnRihGJUYoRilGKEYo sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; LCZJInhHNiIiIiJGJSEiIg== LCZJInhHNiIiIiIiIiNGJQ== IiIi u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCZJInhHNiIiIiIiIiZGJQ== LCgqJClJInhHNiIiIiMiIiJGKCIiKCEiIiomIiImRihGJUYoRig= LCYqJiIkXSUiIiJJInhHNiJGJSEiIiIldl1GJQ== IiNE c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCYqJiIjPSIiIkkieEc2IkYlISIiIiQuI0Yl LCYqJClJInhHNiIiIiMiIiJGKEYlISIi LCZJInhHNiIhIiIiIiJGJg== sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; LCZJInhHNiIhIiIiIiMiIiI= IiIi IiIi u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCZJInhHNiIiIiIiI0lGJQ== LCgqJClJInhHNiIiIiMiIiJGKCIjVkYoKiYiIz9GKEYlRighIiI= LCYqJiIkRCIiIiJJInhHNiJGJUYlIiVdUEYl IiREIg== c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCZJInhHNiIiIiIiI0lGJQ== LCQqJiIiIyIiIilJInhHNiJGJEYlISIi LCQqJiIiIyIiIkkieEc2IkYlRiU= sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; IiIi ISIj IiIh u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCZJInhHNiIiIiIiI0lGJQ== LCgqJClJInhHNiIiIiMiIiJGKCIkbyJGKComIiM/RihGJUYoISIi IiIh IiREJw==
<Text-field style="Heading 2" layout="Heading 2">E 6.6. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=5; m:=p; a:=x^4+1; a mod p; u:=x^2+2; w:=x^2-2; expand(u*w) mod p; e:=expand(a-u*w); IiIm IiIm LCYqJClJInhHNiIiIiUiIiJGKEYoRig= LCYqJClJInhHNiIiIiUiIiJGKEYoRig= LCYqJClJInhHNiIiIiMiIiJGKEYnRig= LCYqJClJInhHNiIiIiMiIiJGKEYnISIi LCYqJClJInhHNiIiIiUiIiJGKEYoRig= IiIm Gcdex(u,w,x,'s','t') mod p; s; t; IiIi ISIi IiIi c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; IiIi ISIi IiIi sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; ISIi IiIh IiIi u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCYqJClJInhHNiIiIiMiIiJGKCIiKEYo LCYqJClJInhHNiIiIiMiIiJGKCIiKCEiIg== IiNd IiNE c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; IiIj ISIj IiIj sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; ISIj IiIh IiIj u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCYqJClJInhHNiIiIiMiIiJGKCIjZEYo LCYqJClJInhHNiIiIiMiIiJGKCIjZCEiIg== IiVdSw== IiREIg== c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; IiNF ISIi IiIi sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; ISIi IiIh IiIi u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCYqJClJInhHNiIiIiMiIiJGKCIkIz1GKA== LCYqJClJInhHNiIiIiMiIiJGKCIkIz0hIiI= IiZESiQ= IiREJw== c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; IiNg IiIj ISIj sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; IiIj IiIh ISIj u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCYqJClJInhHNiIiIiMiIiJGKCIlbzUhIiI= LCYqJClJInhHNiIiIiMiIiJGKCIlbzVGKA== IihEMTki IiVESg==
<Text-field style="Heading 2" layout="Heading 2">E 6.7. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=5; m:=p; a:=expand((2*x+5)*(6*x^2-10*x+7)); a mod p; u:=2*x; w:=x^2+2; e:=expand(a-u*w); IiIm IiIm LCoqJiIjNyIiIilJInhHNiIiIiRGJUYlKiYiIzVGJSlGJyIiI0YlRiUqJiIjT0YlRidGJSEiIiIjTkYl LCYqJiIiIyIiIilJInhHNiIiIiRGJUYlRichIiI= LCQqJiIiIyIiIkkieEc2IkYlRiU= LCYqJClJInhHNiIiIiMiIiJGKEYnRig= LCoqJiIjNSIiIilJInhHNiIiIiRGJUYlKiZGJEYlKUYnIiIjRiVGJSomIiNTRiVGJ0YlISIiIiNORiU= Gcdex(u,w,x,'s','t') mod p; s; t; IiIi SSJ4RzYi ISIj c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCoqJiIiIyIiIilJInhHNiIiIiRGJUYlKiZGJEYlKUYnRiRGJUYlKiYiIilGJUYnRiUhIiIiIihGJQ== LCoqJiIiIyIiIilJInhHNiIiIiVGJUYlKiZGJEYlKUYnIiIkRiVGJSomRiRGJSlGJ0YkRiVGJSomRiRGJUYnRiVGJQ== LCoqJClJInhHNiIiIiQiIiJGKCokKUYlIiIjRihGKEYlRihGKEYo sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; LCYqJiIiIyIiIkkieEc2IkYlISIiRiVGKA== LCgqJiIiIyIiIilJInhHNiJGJEYlRiUqJkYkRiVGJ0YlRiVGJCEiIg== LCYqJiIiIyIiIkkieEc2IkYlRiVGJUYl u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCYqJiIjNyIiIkkieEc2IkYlRiUiIiZGJQ== LCgqJClJInhHNiIiIiMiIiJGKCIiJCEiIiomIiM1RihGJUYoRio= LCgqJiIkRCIiIiIpSSJ4RzYiIiIjRiVGJSomIiNdRiVGJ0YlRiVGK0Yl IiNE c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCgqJiIiJiIiIilJInhHNiIiIiNGJUYlKiZGKUYlRidGJUYlRilGJQ== LCYqJiIiIyIiIilJInhHNiJGJEYlRiUqJkYkRiVGJ0YlRiU= LCZJInhHNiIiIiJGJUYl sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; LCYqJiIiIyIiIkkieEc2IkYlRiVGJUYl IiIj IiIi u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCYqJiIjNyIiIkkieEc2IkYlRiUiI0lGJQ== LCgqJClJInhHNiIiIiMiIiJGKCIjQUYoKiYiI1NGKEYlRihGKA== LCgqJiIkKyYiIiIpSSJ4RzYiIiIjRiUhIiIqJiIlKzpGJUYnRiVGKiIkRCdGKg== IiREIg== c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCgqJiIiJSIiIilJInhHNiIiIiNGJSEiIiomIiM3RiVGJ0YlRioiIiZGKg== LCYqJClJInhHNiIiIiQiIiJGKComIiIjRigpRiVGKkYoISIi LCYqJiIiIyIiIilJInhHNiJGJEYlISIiRidGKQ== sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; LCYqJiIiIyIiIkkieEc2IkYlISIiRiVGKA== LCZJInhHNiIiIiIiIiMhIiI= IiIh u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCYqJiIjNyIiIkkieEc2IkYlRiUiI0lGJQ== LCgqJClJInhHNiIiIiMiIiJGKCIkLiIhIiIqJiIkNSNGKEYlRihGKg== LCgqJiIlK0QiIiIpSSJ4RzYiIiIjRiVGJSomIiUrdkYlRidGJUYlIiVESkYl IiREJw== c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCgqJiIiJSIiIilJInhHNiIiIiNGJUYlKiYiIzdGJUYnRiVGJSIiJkYl LCYqJClJInhHNiIiIiQiIiIhIiIqJiIiI0YoKUYlRitGKEYo LCYqJiIiIyIiIilJInhHNiJGJEYlRiVGJ0Yl sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; LCYqJiIiIyIiIkkieEc2IkYlRiVGJUYl LCZJInhHNiIhIiIiIiMiIiI= IiIh u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(a-u*w); m:=m*p; LCYqJiIjNyIiIkkieEc2IkYlRiUiI0lGJQ== LCgqJClJInhHNiIiIiMiIiJGKCIkQSZGKComIiVTNUYoRiVGKEYo LCgqJiImK0QiIiIiKUkieEc2IiIiI0YlISIiKiYiJit2JEYlRidGJUYqIiZEYyJGKg== IiVESg==
<Text-field style="Heading 2" layout="Heading 2">E 6.8. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=5; m:=p; a:=expand((2*x+5)*(6*x^2-10*x+7)); a mod p; u:=2*x; w:=x^2+2; alpha:=lcoeff(a); mu:=lcoeff(u); nu:=lcoeff(w); aa:=alpha*a; u:=alpha*u/mu mod m; w:=alpha*w/nu mod m; e:=expand(aa-u*w); IiIm IiIm LCoqJiIjNyIiIilJInhHNiIiIiRGJUYlKiYiIzVGJSlGJyIiI0YlRiUqJiIjT0YlRidGJSEiIiIjTkYl LCYqJiIiIyIiIilJInhHNiIiIiRGJUYlRichIiI= LCQqJiIiIyIiIkkieEc2IkYlRiU= LCYqJClJInhHNiIiIiMiIiJGKEYnRig= IiM3 IiIj IiIi LCoqJiIkVyIiIiIpSSJ4RzYiIiIkRiVGJSomIiQ/IkYlKUYnIiIjRiVGJSomIiRLJUYlRidGJSEiIiIkPyVGJQ== LCQqJiIiIyIiIkkieEc2IkYlRiU= LCYqJiIiIyIiIilJInhHNiJGJEYlRiVGJSEiIg== LCoqJiIkUyIiIiIpSSJ4RzYiIiIkRiVGJSomIiQ/IkYlKUYnIiIjRiVGJSomIiRJJUYlRidGJSEiIiIkPyVGJQ== Gcdex(u,w,x,'s','t') mod p; s; t; IiIi SSJ4RzYi ISIi c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCoqJiIjRyIiIilJInhHNiIiIiRGJUYlKiYiI0NGJSlGJyIiI0YlRiUqJiIjJylGJUYnRiUhIiIiIyUpRiU= LCoqJiIiIyIiIilJInhHNiIiIiVGJSEiIiokKUYnIiIkRiVGKiokKUYnRiRGJUYqRidGKg== LCoqJiIiIyIiIilJInhHNiIiIiRGJUYlKiQpRidGJEYlRiVGJ0YlRiVGJQ== sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; LCZJInhHNiIiIiJGJSEiIg== LCgqJClJInhHNiIiIiMiIiIhIiIqJkYnRihGJUYoRihGKEYp LCZJInhHNiIhIiIiIiJGJg== u:=expand(u+tau*m); w:=expand(w+sigma*m); m:=m*p; mu:=lcoeff(u); nu:=lcoeff(w); u:=alpha*u/mu mod m; w:=alpha*w/nu mod m; e:=expand(aa-u*w); LCYqJiIiJCIiIkkieEc2IkYlISIiIiImRiU= LCgqJiIiIyIiIilJInhHNiJGJEYlRiUiIichIiIqJiIiJkYlRidGJUYl IiNE ISIk IiIj LCYqJiIjNyIiIkkieEc2IkYlRiUiIiZGJQ== LCgqJiIjNyIiIilJInhHNiIiIiNGJUYlIiM2ISIiKiYiIiZGJUYnRiVGJQ== LCYqJiIkRCQiIiJJInhHNiJGJSEiIiIkdiVGJQ== c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; LCYqJiIjOCIiIkkieEc2IkYlISIiIiM+RiU= LCYqJiIiIyIiIilJInhHNiJGJEYlRiVGJyEiIg== LCYqJiIiIyIiIkkieEc2IkYlISIiRiVGJQ== sigma:=Rem(sigma,w,x,'q') mod p; q; tau:=expand(tau+q*u) mod p; LCZJInhHNiIhIiIiIiJGJg== IiIi IiIi u:=expand(u+tau*m); w:=expand(w+sigma*m); m:=m*p; mu:=lcoeff(u); nu:=lcoeff(w); u:=alpha*u/mu mod m; w:=alpha*w/nu mod m; e:=expand(aa-u*w); LCYqJiIjNyIiIkkieEc2IkYlRiUiI0lGJQ== LCgqJiIjNyIiIilJInhHNiIiIiNGJUYlIiM5RiUqJiIjP0YlRidGJSEiIg== IiREIg== IiM3 IiM3 LCYqJiIjNyIiIkkieEc2IkYlRiUiI0lGJQ== LCgqJiIjNyIiIilJInhHNiIiIiNGJUYlIiM5RiUqJiIjP0YlRidGJSEiIg== IiIh mu:=igcd(coeffs(u)); u:=u/mu; nu:=igcd(coeffs(w)); w:=w/nu; IiIn LCYqJiIiIyIiIkkieEc2IkYlRiUiIiZGJQ== IiIj LCgqJiIiJyIiIilJInhHNiIiIiNGJUYlKiYiIzVGJUYnRiUhIiIiIihGJQ==
<Text-field style="Heading 2" layout="Heading 2">A 6.1. Algoritmus. </Text-field> replace_lc:=proc(a,x,alpha) local aa,aalpha,t; aa:=expand(a); aalpha:=lcoeff(aa,x,'t'); aa:=expand(aa-aalpha*t+alpha*t); end; Zio2JUkiYUc2IkkieEdGJUkmYWxwaGFHRiU2JUkjYWFHRiVJJ2FhbHBoYUdGJUkidEdGJUYlRiVDJT5GKS1JJ2V4cGFuZEclKnByb3RlY3RlZEc2I0YkPkYqLUknbGNvZWZmR0YwNiVGKUYmLkYrPkYpLUYvNiMsKEYpIiIiKiZGKkY7RitGOyEiIiomRidGO0YrRjtGO0YlRiVGJQ== UnivariateHensel:=proc(a,u1,w1,x,p,B,gamma) local aa,alpha,e,u,uu,w,ww,m,s,t,q,c,sigma,tau; aa:=expand(a); alpha:=lcoeff(aa); aa:=gamma*aa; uu:=expand(u1); uu:=uu/lcoeff(uu)*gamma mod p; ww:=expand(w1); ww:=ww/lcoeff(ww)*alpha mod p; Gcdex(uu,ww,x,'s','t') mod p; u:=replace_lc(uu,x,gamma); w:=replace_lc(ww,x,alpha); e:=expand(aa-u*w); m:=p; while e<>0 and m<2*B*gamma do c:=e/m; sigma:=expand(s*c) mod p; tau:=expand(t*c) mod p; sigma:=Rem(sigma,ww,x,'q') mod p; tau:=expand(tau+q*uu) mod p; u:=expand(u+tau*m); w:=expand(w+sigma*m); e:=expand(aa-u*w); m:=m*p; od; if e=0 then u:=u/igcd(coeffs(u)); w:=w/igcd(coeffs(w)); [u,w]; else FAIL fi; end; 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 6.9. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> debug(UnivariateHensel); debug(replace_lc); STFVbml2YXJpYXRlSGVuc2VsRzYi SStyZXBsYWNlX2xjRzYi UnivariateHensel(a,2*x,2*x^2-1,x,5,10000,2); {--> enter UnivariateHensel, args = 12*x^3+10*x^2-36*x+35, 2*x, 2*x^2-1, x, 5, 10000, 2 LCoqJiIjNyIiIilJInhHNiIiIiRGJUYlKiYiIzVGJSlGJyIiI0YlRiUqJiIjT0YlRidGJSEiIiIjTkYl IiM3 LCoqJiIjQyIiIilJInhHNiIiIiRGJUYlKiYiIz9GJSlGJyIiI0YlRiUqJiIjc0YlRidGJSEiIiIjcUYl LCQqJiIiIyIiIkkieEc2IkYlRiU= LCQqJiIiIyIiIkkieEc2IkYlRiU= LCYqJiIiIyIiIilJInhHNiJGJEYlRiVGJSEiIg== LCYqJiIiIyIiIilJInhHNiJGJEYlRiVGJSEiIg== IiIi {--> enter replace_lc, args = 2*x, x, 2 LCQqJiIiIyIiIkkieEc2IkYlRiU= IiIj LCQqJiIiIyIiIkkieEc2IkYlRiU= <-- exit replace_lc (now in UnivariateHensel) = 2*x} LCQqJiIiIyIiIkkieEc2IkYlRiU= {--> enter replace_lc, args = 2*x^2-1, x, 12 LCYqJiIiIyIiIilJInhHNiJGJEYlRiVGJSEiIg== IiIj LCYqJiIjNyIiIilJInhHNiIiIiNGJUYlRiUhIiI= <-- exit replace_lc (now in UnivariateHensel) = 12*x^2-1} LCYqJiIjNyIiIilJInhHNiIiIiNGJUYlRiUhIiI= LCgqJiIjPyIiIilJInhHNiIiIiNGJUYlKiYiI3FGJUYnRiUhIiJGK0Yl IiIm LCgqJiIiJSIiIilJInhHNiIiIiNGJUYlKiYiIzlGJUYnRiUhIiJGK0Yl LCgqJClJInhHNiIiIiQiIiIhIiIqJClGJSIiI0YoRihGJUYp LCgqJClJInhHNiIiIiMiIiJGKEYlISIiRihGKA== LCZJInhHNiIiIiIiIiMhIiI= IiIi LCYqJiIiIyIiIkkieEc2IkYlRiUiIiZGJQ== LCgqJiIjNyIiIilJInhHNiIiIiNGJUYlIiM2ISIiKiYiIiZGJUYnRiVGJQ== LCgqJiIjXSIiIilJInhHNiIiIiNGJSEiIiomIiN2RiVGJ0YlRioiJEQiRiU= IiNE LCgqJiIiIyIiIilJInhHNiJGJEYlISIiKiYiIiRGJUYnRiVGKSIiJkYl LCYqJiIiIyIiIilJInhHNiIiIiRGJSEiIiomRiRGJSlGJ0YkRiVGJQ== LCYqJiIiIyIiIilJInhHNiJGJEYlRiUqJkYkRiVGJ0YlISIi LCZJInhHNiIhIiIiIiJGJg== IiIh LCYqJiIiIyIiIkkieEc2IkYlRiUiIiZGJQ== LCgqJiIjNyIiIilJInhHNiIiIiNGJUYlIiM5RiUqJiIjP0YlRidGJSEiIg== IiIh IiREIg== LCYqJiIiIyIiIkkieEc2IkYlRiUiIiZGJQ== LCgqJiIiJyIiIilJInhHNiIiIiNGJUYlKiYiIzVGJUYnRiUhIiIiIihGJQ== NyQsJiomIiIjIiIiSSJ4RzYiRiZGJiIiJkYmLCgqJiIiJ0YmKUYnRiVGJkYmKiYiIzVGJkYnRiYhIiIiIihGJg== <-- exit UnivariateHensel (now at top level) = [2*x+5, 6*x^2-10*x+7]} NyQsJiomIiIjIiIiSSJ4RzYiRiZGJiIiJkYmLCgqJiIiJ0YmKUYnRiVGJkYmKiYiIzVGJkYnRiYhIiIiIihGJg== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 6.10. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=5; l:=1; a:=x^2*y^4*z-x*y^9*z^2+x*y*z^3+2*x-y^6*z^4-2*y^5*z; subs(y=1,z=1,a) mod p^l; IiIm IiIi LC4qKClJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRihJInpHRiZGKEYoKihGJUYoKUYqIiIqRigpRixGJ0YoISIiKihGJUYoRipGKClGLCIiJEYoRigqJkYnRihGJUYoRigqJilGKiIiJ0YoKUYsRitGKEYxKihGJ0YoKUYqIiImRihGLEYoRjE= LCgqJClJInhHNiIiIiMiIiJGKComRidGKEYlRihGKEYnRig= u[1]:=x-2; w[1]:=x-1; expand(u[1]*w[1]) mod p^l; LCZJInhHNiIiIiIiIiMhIiI= LCZJInhHNiIiIiJGJSEiIg== LCgqJClJInhHNiIiIiMiIiJGKComRidGKEYlRihGKEYnRig= aa:=expand(subs(y=Y+1,z=Z+1,a)) mod p^l; 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 collect(aa,[Y,Z],`distributed`): aa:=sort(%,[Y,Z],tdeg); 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 aa:=subs(Y=y-1,Z=z-1,aa) mod p^l; u[7]:=(x-2)+(-x+1)*(y-1)+(x-2)*(z-1)+x*(y-1)^2+(-x-2)*(y-1)*(z-1)+(-2)*(z-1)^2+(-x)*(y-1)^3+x*(y-1)^2*(z-1)+(-2)*(y-1)*(z-1)^2+(z-1)^3+x*(y-1)^4+(-x)*(y-1)^3*(z-1)+(y-1)*(z-1)^3+x*(y-1)^4*(z-1) mod p^l; w[7]:=(x-1)+(-1)*(z-1)+(-1)*(y-1)^5+(-1)*(y-1)^5*(z-1) mod p^l; 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 LEBJInhHNiIiIiIiIiMhIiIqJiwmRiNGJ0YlRiVGJSwmSSJ5R0YkRiVGJUYnRiVGJSomLCZGI0YlRiZGJ0YlLCZJInpHRiRGJUYlRidGJUYlKiZGI0YlKUYqRiZGJUYlKigsJkYjRidGJkYnRiVGKkYlRi5GJUYlKiZGJkYlKUYuRiZGJUYnKiZGI0YlKUYqIiIkRiVGJyooRiNGJUYxRiVGLkYlRiUqKEYmRiVGKkYlRjVGJUYnKiQpRi5GOEYlRiUqJkYjRiUpRioiIiVGJUYlKihGI0YlRjdGJUYuRiVGJyomRipGJUY8RiVGJSooRiNGJUY+RiVGLkYlRiU= LCpJInhHNiIiIiJJInpHRiQhIiIqJCksJkkieUdGJEYlRiVGJyIiJkYlRicqJkYpRiUsJkYmRiVGJUYnRiVGJw== expand(u[7]) mod p^l; expand(w[7]) mod p^l; expand(aa-u[7]*w[7]) mod p^l; LCgiIiMiIiIqKEkieEc2IkYkKUkieUdGJyIiJUYkSSJ6R0YnRiRGJComRilGJClGKyIiJEYkRiQ= LCZJInhHNiIiIiIqJilJInlHRiQiIiZGJUkiekdGJEYlISIi IiIh
<Text-field style="Heading 2" layout="Heading 2">A 6.2. Algoritmus. </Text-field> MultivariateDiophant:=proc(a,c,E,d,p,k) local sigma,r,nu,i,A,aa,b,cc,EE,e,monom,m,x,y,ee,cm,ds,alpha; r:=nops(a); nu:=nops(E); if nu>1 then x:=op(1,E[nu]); alpha:=op(2,E[nu]); A:=mul(a[i],i=1..r); for i to r do b[i]:=A/a[i] od; aa:=subs(E[nu],a); cc:=subs(E[nu],c); EE:=E[1..nu-1]; sigma:=MultivariateDiophant(aa,cc,EE,d,p,k); e:=mods(expand(c-add(sigma[i]*b[i],i=1..r)),p^k); monom:=1; for m to d while e<>0 do monom:=monom*(x-alpha); ee:=diff(e,[x$m]); cm:=subs(x=alpha,ee)/m!; if cm<>0 then ds:=MultivariateDiophant(aa,cm,EE,d,p,k); for i to r do sigma[i]:=expand(sigma[i]+ds[i]*monom) od; e:=mods(expand(e-add(ds[i]*monom*b[i],i=1..r)),p^k); fi; od; else x:=E[1]; sigma:=[0$i=1..r]; for m from 0 to d do cm:=coeff(c,x,m); if cm<>0 then ds:=UnivariateDiophant(a,x,m,p,k); for i to r do sigma[i]:=expand(sigma[i]+ds[i]*cm) od; fi; od; fi; map((x,y)->mods(x,y),sigma,p^k); end; 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
<Text-field style="Heading 2" layout="Heading 2">A 6.3. Algoritmus. </Text-field> UnivariateDiophant:=proc(a,x,m,p,k) local i,sigma,r,s,R,q; r:=nops(a); if r>2 then s:=MultiTermEEAlift(a,x,p,k); R:=[]; for i to r do R:=[op(R),mods(rem(x^m*s[i],a[i],x),p^k)] od; else s:=EEAlift(a[2],a[1],x,p,k); q:=mods(quo(x^m*s[1],a[1],x),p^k); R:=[mods(expand(x^m*s[1]-q*a[1]),p^k), mods(expand(x^m*s[2]+q*a[2]),p^k)]; fi; R; end; 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 MultiTermEEAlift:=proc(a,x,p,k) local i,r,s,beta,sigma; r:=nops(a); s:=[0$i=1..r]; s[r-1]:=a[r]; for i from r-2 by -1 to 1 do s[i]:=expand(a[i+1]*s[i+1]) od; beta:=1; for i to r-1 do sigma:=MultivariateDiophant([s[i],a[i]],beta,[x],0,p,k); beta:=sigma[1]; s[i]:=sigma[2]; od; s[r]:=beta; s; end; Zio2JkkiYUc2IkkieEdGJUkicEdGJUkia0dGJTYnSSJpR0YlSSJyR0YlSSJzR0YlSSViZXRhR0YlSSZzaWdtYUdGJUYlRiVDKj5GKy1JJW5vcHNHJSpwcm90ZWN0ZWRHNiNGJD5GLDcjLUkiJEdGMzYkIiIhL0YqOyIiIkYrPiZGLDYjLCZGK0Y9Rj0hIiImRiQ2I0YrPyhGKiwmRitGPSIiI0ZCRkJGPUkldHJ1ZUdGMz4mRiw2I0YqLUknZXhwYW5kR0YzNiMqJiZGJDYjLCZGKkY9Rj1GPUY9JkYsRlFGPT5GLUY9PyhGKkY9Rj1GQUZIQyU+Ri4tSTVNdWx0aXZhcmlhdGVEaW9waGFudEdGJTYoNyRGSiZGJEZLRi03I0YmRjpGJ0YoPkYtJkYuNiNGPT5GSiZGLjYjRkc+JkYsRkRGLUYsRiVGJUYl EEAlift:=proc(a,b,x,p,k) local ap,bp,s,t,sp,tp,i,m,e,c,q,sigma,tau; ap:=mods(a,p); bp:=mods(b,p); mods(Gcdex(ap,bp,x,'s','t'),p); sp:=mods(s,p); tp:=mods(t,p); m:=p; for i to k-1 do e:=expand(1-s*a-t*b); c:=mods(e/m,p); sigma:=mods(expand(sp*c),p); tau:=mods(expand(tp*c),p); q:=mods(Quo(sigma,bp,x),p); sigma:=mods(expand(sigma-q*bp),p); tau:=mods(expand(tau+q*ap),p); s:=expand(s+sigma*m); t:=expand(t+tau*m); m:=m*p; od; [s,t]; end; 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 LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">A 6.4. Algoritmus. </Text-field> MultivariateHensel:=proc(a,E,p,l,u,lcU) local nu,A,i,x,alpha,U,UU,n,monom,maxdeg,aa,e,ee,co,oco,t,xx,m,j,c,dU; aa:=expand(a); nu:=nops(E); A:=[0$i=1..nu]; n:=nops(u); A[nu]:=aa; maxdeg:=-1; for i from nu by -1 to 2 do x:=op(1,E[i]); alpha:=op(2,E[i]); A[i-1]:=subs(E[i],A[i]); if degree(a,x)>maxdeg then maxdeg:=degree(a,x) fi; od; U:=u; xx:=E[1]; for i from 2 to nu do UU:=U; monom:=1; x:=op(1,E[i]); alpha:=op(2,E[i]); for m to n do if lcU[m]<>1 then co:=mods(subs(E[i+1..nu],lcU[m]),p^l); oco:=lcoeff(collect(U[m],xx),xx,'t'); U[m]:=expand(U[m]-oco*t+co*t); fi; od; e:=expand(A[i]-mul(U[j],j=1..n)); for j to degree(A[i],x) while e<>0 do monom:=monom*(x-alpha); c:=subs(E[i],diff(e,[x$j]))/j!; if c<>0 then dU:=MultivariateDiophant(UU,c,E[1..i-1],maxdeg,p,l); for m to n do U[m]:=mods(expand(U[m]+dU[m]*monom),p^l) od; e:=mods(expand(A[i]-mul(U[m],m=1..n)),p^l); fi; od; od; if a=expand(mul(U[m],m=1..n)) then U else FAIL fi; end; 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 a; factor(a); collect(a,x); coeffs(%,x); gcd(%[1],%[2]); LC4qKClJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRihJInpHRiZGKEYoKihGJUYoKUYqIiIqRigpRixGJ0YoISIiKihGJUYoRipGKClGLCIiJEYoRigqJkYnRihGJUYoRigqJilGKiIiJ0YoKUYsRitGKEYxKihGJ0YoKUYqIiImRihGLEYoRjE= KiYsJkkieEc2IiIiIiomKUkieUdGJSIiJkYmSSJ6R0YlRiYhIiJGJiwoKigpRikiIiVGJkYrRiZGJEYmRiYqJkYpRiYpRisiIiRGJkYmIiIjRiZGJg== LCoqKClJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRihJInpHRiZGKEYoKiYsKComRipGKClGLCIiJEYoRihGJ0YoKiYpRioiIipGKClGLEYnRighIiJGKEYlRihGKComKUYqIiInRigpRixGK0YoRjYqKEYnRigpRioiIiZGKEYsRihGNg== NiUsJiomKUkieUc2IiIiJyIiIilJInpHRiciIiVGKSEiIiooIiIjRikpRiYiIiZGKUYrRilGLSwoKiZGJkYpKUYrIiIkRilGKUYvRikqJilGJiIiKkYpKUYrRi9GKUYtKiYpRiZGLEYpRitGKQ== IiIi E:=[x,z=1,y=1]; subs(E[2..3],a); NyVJInhHNiIvSSJ6R0YkIiIiL0kieUdGJEYn LCgqJClJInhHNiIiIiMiIiJGKComRidGKEYlRihGKCIiJCEiIg== debug(MultivariateHensel); STNNdWx0aXZhcmlhdGVIZW5zZWxHNiI= MultivariateHensel(a,E,5,2,[x-1,x+3],[1,y^4*z]); {--> enter MultivariateHensel, args = x^2*y^4*z-x*y^9*z^2+x*y*z^3+2*x-y^6*z^4-2*y^5*z, [x, z = 1, y = 1], 5, 2, [x-1, x+3], [1, y^4*z] LC4qKClJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRihJInpHRiZGKEYoKihGJUYoKUYqIiIqRigpRixGJ0YoISIiKihGJUYoRipGKClGLCIiJEYoRigqJkYnRihGJUYoRigqJilGKiIiJ0YoKUYsRitGKEYxKihGJ0YoKUYqIiImRihGLEYoRjE= IiIk NyUiIiFGI0Yj IiIj LC4qKClJInhHNiIiIiMiIiIpSSJ5R0YmIiIlRihJInpHRiZGKEYoKihGJUYoKUYqIiIqRigpRixGJ0YoISIiKihGJUYoRipGKClGLCIiJEYoRigqJkYnRihGJUYoRigqJilGKiIiJ0YoKUYsRitGKEYxKihGJ0YoKUYqIiImRihGLEYoRjE= ISIi SSJ5RzYi IiIi LC4qJilJInhHNiIiIiMiIiJJInpHRiZGKEYoKiZGJUYoKUYpRidGKCEiIiomRiVGKClGKSIiJEYoRigqJkYnRihGJUYoRigqJClGKSIiJUYoRiwqJkYnRihGKUYoRiw= IiIq SSJ6RzYi IiIi LCgqJClJInhHNiIiIiMiIiJGKComRidGKEYlRihGKCIiJCEiIg== NyQsJkkieEc2IiIiIkYmISIiLCZGJEYmIiIkRiY= SSJ4RzYi NyQsJkkieEc2IiIiIkYmISIiLCZGJEYmIiIkRiY= IiIi SSJ6RzYi IiIi SSJ6RzYi IiIi LCYiIiQiIiIqJkkiekc2IkYkSSJ4R0YnRiRGJA== LDAqJkkieEc2IiIiIilJInpHRiUiIiNGJiEiIiomRiRGJilGKCIiJEYmRiZGJEYqKiQpRigiIiVGJkYqKiZGKUYmRihGJkYqRi1GJiomRihGJkYkRiZGJg== LCZJInpHNiIiIiJGJSEiIg== LCYqJiIiIyIiIkkieEc2IkYlRiUiIichIiI= NyQhIiIiIiQ= LCZJInhHNiIiIiJJInpHRiQhIiI= LCYqJkkiekc2IiIiIkkieEdGJUYmRiYqJiIiJEYmRiRGJkYm LC4qJkkieEc2IiIiIilJInpHRiUiIiRGJkYmKiYiIiNGJkYkRiZGJiokKUYoIiIlRiYhIiIqJkYrRiZGKEYmRi8qKEYpRiZGKEYmRiRGJkYvKiZGKUYmKUYoRitGJkYm KiQpLCZJInpHNiIiIiJGJyEiIiIiI0Yn LCYqJiIiJCIiIkkieEc2IkYlRiVGJCEiIg== NyQiIiEiIiQ= LCZJInhHNiIiIiJJInpHRiQhIiI= LCoqJkkiekc2IiIiIkkieEdGJUYmRiYqJiIiJEYmRiRGJiEiIiomRilGJilGJCIiI0YmRiZGKUYm LDIqKCIiJCIiIkkieEc2IkYlKUkiekdGJyIiI0YlISIiKiZGJkYlKUYpRiRGJUYlRiZGKyokKUYpIiIlRiVGK0YpRiUqKEYkRiVGKUYlRiZGJUYlKiZGJEYlRihGJUYrKiZGJEYlRi1GJUYl KiQpLCZJInpHNiIiIiJGJyEiIiIiJEYn LCZJInhHNiIiIiJGJSEiIg== NyQiIiEiIiI= LCZJInhHNiIiIiJJInpHRiQhIiI= LCgqJkkiekc2IiIiIkkieEdGJUYmRiYiIiNGJiokKUYkIiIkRiZGJg== IiIh NyQsJkkieEc2IiIiIkkiekdGJSEiIiwoKiZGJ0YmRiRGJkYmIiIjRiYqJClGJyIiJEYmRiY= IiIi SSJ5RzYi IiIi KiYpSSJ5RzYiIiIlIiIiSSJ6R0YlRic= SSJ6RzYi LCgiIiMiIiIqJClJInpHNiIiIiRGJEYkKigpSSJ5R0YoIiIlRiRGJ0YkSSJ4R0YoRiRGJA== LDIqKEkieEc2IiIiIilJInlHRiUiIipGJilJInpHRiUiIiNGJiEiIiooRiRGJkYoRiYpRisiIiRGJkYmKiYpRigiIidGJilGKyIiJUYmRi0qKEYsRiYpRigiIiZGJkYrRiZGLSomRiRGJkYvRiZGLSomRixGJkYrRiZGJiokRjRGJkYmKigpRihGNUYmRipGJkYkRiZGJg== LCZJInlHNiIiIiJGJSEiIg== LCoqKCIiJiIiIkkieEc2IkYlKUkiekdGJyIiI0YlISIiKiZGJkYlKUYpIiIkRiVGJSomIiInRiUpRikiIiVGJUYrKiYiIzVGJUYpRiVGKw== NyQsJComIiImIiIiSSJ6RzYiRiYhIiIqJClGJyIiJEYm LChJInhHNiIiIiIqJiIiJUYlSSJ6R0YkRiVGJSooIiImRiVJInlHRiRGJUYoRiUhIiI= LCgqKClJInlHNiIiIiUiIiJJInpHRiZGKEkieEdGJkYoRigqJkYlRigpRikiIiRGKEYoIiIjRig= LDQqKEkieEc2IiIiIilJInlHRiUiIipGJilJInpHRiUiIiNGJiEiIiomKUYoIiInRiYpRisiIiVGJkYtKihGLEYmKUYoIiImRiZGK0YmRi0qKkYyRiYpRihGMkYmRipGJkYkRiZGLSooRjJGJkYoRiZGMUYmRi0qJiIiKUYmRitGJkYtKipGNUYmRjRGJkYqRiZGJEYmRiYqKEY1RiYpRihGLEYmRjFGJkYmKigiIzVGJkYoRiZGK0YmRiY= KiQpLCZJInlHNiIiIiJGJyEiIiIiI0Yn LCgqKCIjNSIiIkkieEc2IkYlKUkiekdGJyIiI0YlISIiKiZGJEYlKUYpIiIlRiVGKyomIiM/RiVGKUYlRis= NyQsJComIiM1IiIiSSJ6RzYiRiYhIiIiIiE= LCpJInhHNiIiIiIqJiIiJ0YlSSJ6R0YkRiUhIiIqKCIjNUYlSSJ5R0YkRiVGKEYlRikqKEYrRiVGKEYlKUYsIiIjRiVGKQ== LCgqKClJInlHNiIiIiUiIiJJInpHRiZGKEkieEdGJkYoRigqJkYlRigpRikiIiRGKEYoIiIjRig= LDoqKEkieEc2IiIiIilJInlHRiUiIipGJilJInpHRiUiIiNGJiEiIiomKUYoIiInRiYpRisiIiVGJkYtKihGLEYmKUYoIiImRiZGK0YmRi0qKkYwRiYpRihGMkYmRipGJkYkRiZGJiooRjBGJkYoRiZGMUYmRiYqJiIjN0YmRitGJkYmKioiIzVGJkY0RiZGKkYmRiRGJkYmKihGPEYmKUYoRixGJkYxRiZGJiooRjVGJkYoRiZGK0YmRi0qKkY8RiZGKkYmRi9GJkYkRiZGJiooRjxGJkYxRiYpRigiIiRGJkYmKihGNUYmRitGJkY+RiZGLQ== KiQpLCZJInlHNiIiIiJGJyEiIiIiJEYn LCgqKCIkUyMiIiJJInhHNiJGJSlJInpHRiciIiNGJUYlKiYiIzVGJSlGKSIiJUYlISIiKiYiIz9GJUYpRiVGLw== NyQsJComIiM1IiIiSSJ6RzYiRiYhIiIiIiE= LCxJInhHNiIiIiIqJiIiJUYlSSJ6R0YkRiVGJSooIiM1RiVJInlHRiRGJUYoRiVGJSooIiImRiVGKEYlKUYrIiIjRiUhIiIqKEYqRiUpRisiIiRGJUYoRiVGMA== LCgqKClJInlHNiIiIiUiIiJJInpHRiZGKEkieEdGJkYoRigqJkYlRigpRikiIiRGKEYoIiIjRig= LEAqKEkieEc2IiIiIilJInlHRiUiIipGJilJInpHRiUiIiNGJiEiIiomKUYoIiInRiYpRisiIiVGJkYtKihGLEYmKUYoIiImRiZGK0YmRi0qKkYyRiYpRihGMkYmRipGJkYkRiZGLSooRjJGJkYoRiZGMUYmRi0qJiIiKUYmRitGJkYtKioiIzVGJkY0RiZGKkYmRiRGJkYtKihGPEYmKUYoRixGJkYxRiZGLSooRjVGJkYoRiZGK0YmRiYqKkY1RiZGKkYmRi9GJkYkRiZGJiooRjVGJkYxRiYpRigiIiRGJkYmKihGPEYmRitGJkY+RiZGJioqRjxGJkYkRiYpRigiIihGJkYqRiZGJiooRjxGJkY3RiZGMUYmRiYqKEY1RiZGQkYmRitGJkYt KiQpLCZJInlHNiIiIiJGJyEiIiIiJUYn LCgqKCIkWCMiIiJJInhHNiJGJSlJInpHRiciIiNGJUYlKiYiIiZGJSlGKSIiJUYlISIiKiYiIzVGJUYpRiVGLw== NyQsJComIiImIiIiSSJ6RzYiRiYhIiIiIiE= LC5JInhHNiIiIiJJInpHRiQhIiIqKCIiJkYlSSJ5R0YkRiVGJkYlRiUqKCIjNUYlRiZGJSlGKiIiI0YlRicqKEYsRiUpRioiIiRGJUYmRiVGJSooRilGJSlGKiIiJUYlRiZGJUYn LCgqKClJInlHNiIiIiUiIiJJInpHRiZGKEkieEdGJkYoRigqJkYlRigpRikiIiRGKEYoIiIjRig= LEYqKEkieEc2IiIiIilJInlHRiUiIipGJilJInpHRiUiIiNGJiEiIiomKUYoIiInRiYpRisiIiVGJkYtKihGLEYmKUYoIiImRiZGK0YmRi0qKClGKEYyRiZGKkYmRiRGJkYmKiZGKEYmRjFGJkYmKiZGLEYmRitGJkYmKipGNUYmRjRGJkYqRiZGJEYmRi0qKEY1RiYpRihGLEYmRjFGJkYtKigiIzVGJkYoRiZGK0YmRi0qKkY+RiZGKkYmRi9GJkYkRiZGJiooRj5GJkYxRiYpRigiIiRGJkYmKihGNUYmRitGJkY8RiZGLSoqRj5GJkYkRiYpRigiIihGJkYqRiZGLSooRj5GJkY3RiZGMUYmRi0qKEY1RiZGQUYmRitGJkYmKipGNUYmRiRGJilGKCIiKUYmRipGJkYmKihGNUYmRjRGJkYxRiZGJiooRj5GJkY3RiZGK0YmRiY= KiQpLCZJInlHNiIiIiJGJyEiIiIiJkYn LCgqJkkieEc2IiIiIilJInpHRiUiIiNGJiEiIiokKUYoIiIlRiZGKiomRilGJkYoRiZGKg== NyQsJEkiekc2IiEiIiIiIQ== LCZJInhHNiIiIiIqJilJInlHRiQiIiZGJUkiekdGJEYlISIi LCgqKClJInlHNiIiIiUiIiJJInpHRiZGKEkieEdGJkYoRigqJkYlRigpRikiIiRGKEYoIiIjRig= IiIh NyQsJkkieEc2IiIiIiomKUkieUdGJSIiJkYmSSJ6R0YlRiYhIiIsKCooKUYpIiIlRiZGK0YmRiRGJkYmKiZGKUYmKUYrIiIkRiZGJiIiI0Ym <-- exit MultivariateHensel (now at top level) = [x-y^5*z, y^4*z*x+y*z^3+2]} NyQsJkkieEc2IiIiIiomKUkieUdGJSIiJkYmSSJ6R0YlRiYhIiIsKCooKUYpIiIlRiZGK0YmRiRGJkYmKiZGKUYmKUYrIiIkRiZGJiIiI0Ym
<Text-field style="Heading 1" layout="Heading 1">7. Legnagyobb k<Font encoding="UTF-8">\303\266</Font>z<Font encoding="UTF-8">\303\266</Font>s oszt<Font encoding="UTF-8">\303\263</Font></Text-field>
<Text-field style="Heading 1" layout="Heading 1">8. Faktoriz<Font encoding="UTF-8">\303\241</Font>l<Font encoding="UTF-8">\303\241</Font>s</Text-field>
<Text-field style="Heading 1" layout="Heading 1">9. Egyenletrendszerek</Text-field>
<Text-field style="Heading 1" layout="Heading 1">10. Gr<Font encoding="UTF-8">\303\266bner-b\303\241zisok</Font></Text-field>
<Text-field style="Heading 1" layout="Heading 1">11. Racion<Font encoding="UTF-8">\303\241</Font>lis t<Font encoding="UTF-8">\303\266</Font>rtf<Font encoding="UTF-8">\303\274</Font>ggv<Font encoding="UTF-8">\303\251</Font>nyek integr<Font encoding="UTF-8">\303\241</Font>l<Font encoding="UTF-8">\303\241</Font>sa</Text-field>
<Text-field style="Heading 1" layout="Heading 1">12. A Risch-algoritmus.</Text-field>
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn