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> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 1" layout="Heading 1">2. Algebrai alapok</Text-field>
<Text-field style="Heading 2" layout="Heading 2">2.1. </Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">2.2.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<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>
<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 2" layout="Heading 2">E 10.1. Lexikografikus rendez<Font encoding="UTF-8">\303\251s.</Font></Text-field> p:=x+y+z+x^2+z^3+x*y+y^2; LDBJInhHNiIiIiJJInlHRiRGJUkiekdGJEYlKiQpRiMiIiNGJUYlKiQpRiciIiRGJUYlKiZGI0YlRiZGJUYlKiQpRiZGKkYlRiU= sort(p,[x,y,z],plex); LDAqJClJInhHNiIiIiMiIiJGKComRiVGKEkieUdGJkYoRihGJUYoKiQpRipGJ0YoRihGKkYoKiQpSSJ6R0YmIiIkRihGKEYvRig= sort(p,[z,y,x],plex); LDAqJClJInpHNiIiIiQiIiJGKEYlRigqJClJInlHRiYiIiNGKEYoKiZGK0YoSSJ4R0YmRihGKEYrRigqJClGLkYsRihGKEYuRig=
<Text-field style="Heading 2" layout="Heading 2">E 10.2. Foksz<Font encoding="UTF-8">\303\241m rendez\303\251s.</Font></Text-field> sort(p,[x,y,z]); LDAqJClJInpHNiIiIiQiIiJGKCokKUkieEdGJiIiI0YoRigqJkYrRihJInlHRiZGKEYoKiQpRi5GLEYoRihGK0YoRi5GKEYlRig= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.3. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=-2*x^2*y*z+x^2*y^2+x^2*z^2+x^2*y+2*x*y^2*z^2-3*x*y*z^3-x*y+y*z+z^2+5; LDYqKiIiIyIiIilJInhHNiJGJEYlSSJ5R0YoRiVJInpHRihGJSEiIiomRiZGJSlGKUYkRiVGJSomRiZGJSlGKkYkRiVGJSomRiZGJUYpRiVGJSoqRiRGJUYnRiVGLUYlRi9GJUYlKioiIiRGJUYnRiVGKUYlKUYqRjNGJUYrKiZGJ0YlRilGJUYrKiZGKUYlRipGJUYlKiRGL0YlRiUiIiZGJQ== sort(p,[x,y,z]); LDYqKiIiIyIiIkkieEc2IkYlKUkieUdGJ0YkRiUpSSJ6R0YnRiRGJUYlKioiIiRGJUYmRiVGKUYlKUYrRi1GJSEiIiomKUYmRiRGJUYoRiVGJSoqRiRGJUYxRiVGKUYlRitGJUYvKiZGMUYlRipGJUYlKiZGMUYlRilGJUYlKiZGJkYlRilGJUYvKiZGKUYlRitGJUYlKiRGKkYlRiUiIiZGJQ== op(1,p); lcoeff(%,[x,y,z],'t'); t; LCQqKiIiIyIiIkkieEc2IkYlKUkieUdGJ0YkRiUpSSJ6R0YnRiRGJUYl IiIj KihJInhHNiIiIiIpSSJ5R0YkIiIjRiUpSSJ6R0YkRihGJQ== collect(p,[x,y]); LCwqJiwoKiQpSSJ5RzYiIiIjIiIiRioqJiwmKiZGKUYqSSJ6R0YoRiohIiJGKkYqRipGJ0YqRioqJClGLkYpRipGKkYqKUkieEdGKEYpRipGKiomLCYqKEYpRipGJkYqRjFGKkYqKiYsJkYqRi8qJiIiJEYqKUYuRjpGKkYvRipGJ0YqRipGKkYzRipGKkYwRioiIiZGKiomRidGKkYuRipGKg== pp:=collect(p,[x,y],`distributed`); LDIqJClJInpHNiIiIiMiIiJGKCIiJkYoKipGJ0YoSSJ4R0YmRigpSSJ5R0YmRidGKEYkRihGKCooLCZGKCEiIiomIiIkRigpRiVGMkYoRjBGKEYrRihGLUYoRigqJilGK0YnRihGLEYoRigqKCwmKiZGJ0YoRiVGKEYwRihGKEYoRjVGKEYtRihGKComRi1GKEYlRihGKComRjVGKEYkRihGKA== sort(pp,[x,y]); LDIqJilJInhHNiIiIiMiIiIpSSJ5R0YmRidGKEYoKigsJiomRidGKEkiekdGJkYoISIiRihGKEYoRiRGKEYqRihGKCoqRidGKClGLkYnRihGJUYoRilGKEYoKiZGMUYoRiRGKEYoKigsJkYoRi8qJiIiJEYoKUYuRjZGKEYvRihGJUYoRipGKEYoKiZGLkYoRipGKEYoKiRGMUYoRigiIiZGKA== op(1,p); lcoeff(%,[x,y],'t'); t; LCQqKiIiIyIiIilJInpHNiJGJEYlSSJ4R0YoRiUpSSJ5R0YoRiRGJUYl LCQqJiIiIyIiIilJInpHNiJGJEYlRiU= KiYpSSJ5RzYiIiIjIiIiSSJ4R0YlRic= with(Groebner); 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 LeadingTerm(p,tdeg(x,y,z)); %[1]*%[2]; NiQiIiMqKEkieEc2IiIiIilJInlHRiZGI0YnKUkiekdGJkYjRic= LCQqKiIiIyIiIkkieEc2IkYlKUkieUdGJ0YkRiUpSSJ6R0YnRiRGJUYl LeadingMonomial(p,tdeg(x,y,z)); KihJInhHNiIiIiIpSSJ5R0YkIiIjRiUpSSJ6R0YkRihGJQ== LeadingCoefficient(p,tdeg(x,y,z)); IiIj LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.4. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=6*x^4+13*x^3-6*x+1; q:=3*x^2+5*x-1; LCoqJiIiJyIiIilJInhHNiIiIiVGJUYlKiYiIzhGJSlGJyIiJEYlRiUqJkYkRiVGJ0YlISIiRiVGJQ== LCgqJiIiJCIiIilJInhHNiIiIiNGJUYlKiYiIiZGJUYnRiVGJUYlISIi p-2*x^2*q; LCwqJiIiJyIiIilJInhHNiIiIiVGJUYlKiYiIzhGJSlGJyIiJEYlRiUqJkYkRiVGJ0YlISIiRiVGJSooIiIjRiUpRidGMUYlLCgqJkYtRiVGMkYlRiUqJiIiJkYlRidGJUYlRiVGL0YlRi8= expand(p-2*x^2*q); LCoqJiIiJCIiIilJInhHNiJGJEYlRiUqJiIiJ0YlRidGJSEiIkYlRiUqJiIiI0YlKUYnRi1GJUYl expand(p-13/3*x*q); LCoqJiIiJyIiIilJInhHNiIiIiVGJUYlKiYjIiImIiIkRiVGJ0YlISIiRiVGJSomIyIjbEYtRiUpRiciIiNGJUYu LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.5. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=2*y^2*z-x*z^2; q1:=7*y^2+y*z-4; q2:=2*y*z-3*x+1; LCYqKCIiIyIiIilJInlHNiJGJEYlSSJ6R0YoRiVGJSomSSJ4R0YoRiUpRilGJEYlISIi LCgqJiIiKCIiIilJInlHNiIiIiNGJUYlKiZGJ0YlSSJ6R0YoRiVGJSIiJSEiIg== LCgqKCIiIyIiIkkieUc2IkYlSSJ6R0YnRiVGJSomIiIkRiVJInhHRidGJSEiIkYlRiU= sort(p,[x,y,z]); sort(q1,[x,y,z]); sort(q2,[x,y,z]); LCYqJkkieEc2IiIiIilJInpHRiUiIiNGJiEiIiooRilGJilJInlHRiVGKUYmRihGJkYm LCgqJiIiKCIiIilJInlHNiIiIiNGJUYlKiZGJ0YlSSJ6R0YoRiVGJSIiJSEiIg== LCgqKCIiIyIiIkkieUc2IkYlSSJ6R0YnRiVGJSomIiIkRiVJInhHRidGJSEiIkYlRiU= expand(p-2/7*z*q1); expand(p-y*q2); LCgqJkkieEc2IiIiIilJInpHRiUiIiNGJiEiIiooI0YpIiIoRiZJInlHRiVGJkYnRiZGKiomIyIiKUYtRiZGKEYmRiY= LCgqJkkieEc2IiIiIilJInpHRiUiIiNGJiEiIiooIiIkRiZGJEYmSSJ5R0YlRiZGJkYtRio= expand(%+1/7*q2); LC4qJkkieEc2IiIiIilJInpHRiUiIiNGJiEiIiooIiIkRiZGJEYmSSJ5R0YlRiZGJkYtRioqKCNGKSIiKEYmRi1GJkYoRiZGJiomI0YsRjBGJkYkRiZGKiNGJkYwRiY= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">A 10.1. Algoritmus.</Text-field> reducers:=proc(p,Q,ord) local phmonom,qhmonom,q,R; R:=[]; phmonom:=LeadingMonomial(p,ord); for q in Q do if q<>0 then qhmonom:=LeadingMonomial(q,ord); if divide(phmonom,qhmonom) then R:=[op(R),q]; fi; fi; od; R; end; Zio2JUkicEc2IkkiUUdGJUkkb3JkR0YlNiZJKHBobW9ub21HRiVJKHFobW9ub21HRiVJInFHRiVJIlJHRiVGJUYlQyY+Riw3Ij5GKS1fSSlHcm9lYm5lckc2JCUqcHJvdGVjdGVkR0koX3N5c2xpYkdGJUkwTGVhZGluZ01vbm9taWFsR0YlNiRGJEYnPyZGK0YmSSV0cnVlR0Y1QCQwRisiIiFDJD5GKi1GMjYkRitGJ0AkLUknZGl2aWRlR0Y1NiRGKUYqPkYsNyQtSSNvcEdGNTYjRixGK0YsRiVGJUYl Q:=[q1,q2]; QQ:=reducers(p,Q,tdeg(x,y,z)); NyQsKComIiIoIiIiKUkieUc2IiIiI0YmRiYqJkYoRiZJInpHRilGJkYmIiIlISIiLCgqKEYqRiZGKEYmRixGJkYmKiYiIiRGJkkieEdGKUYmRi5GJkYm NyQsKComIiIoIiIiKUkieUc2IiIiI0YmRiYqJkYoRiZJInpHRilGJkYmIiIlISIiLCgqKEYqRiZGKEYmRixGJkYmKiYiIiRGJkkieEdGKUYmRi5GJkYm reducer:=proc(p,Q,ord) Q[1] end; Zio2JUkicEc2IkkiUUdGJUkkb3JkR0YlRiVGJUYlJkYmNiMiIiJGJUYlRiU= reducer(p,QQ,tdeg(x,y,z)); LCgqJiIiKCIiIilJInlHNiIiIiNGJUYlKiZGJ0YlSSJ6R0YoRiVGJSIiJSEiIg== myreduce:=proc(p,Q,ord) local QQ,q,pp,ppp,lpp,lq,t; pp:=p; ppp:=0; while pp<>0 do QQ:=reducers(pp,Q,ord); if QQ<>[] then q:=reducer(p,QQ,ord); t:=LeadingTerm(pp,ord); lpp:=t[1]*t[2]; t:=LeadingTerm(q,ord); lq:=t[1]*t[2]; pp:=expand(pp-lpp*q/lq); else LeadingTerm(pp,ord); lpp:=%[1]*%[2]; ppp:=ppp+lpp; pp:=pp-lpp; fi; od; ppp; end; 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 myreduce(p,Q,tdeg(x,y,z)); LCgqJkkieEc2IiIiIilJInpHRiUiIiNGJiEiIiooIyIiJCIiKEYmRihGJkYkRiZGKiomIyIiKkYuRiZGKEYmRiY= LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.6. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> p:=3*x^3*y+2*x^2*y^2-3*x*y+5*x; q1:=x^2*y+5*x^2+y^2; q2:=7*x*y^2-2*y^3+1; LCoqKCIiJCIiIilJInhHNiJGJEYlSSJ5R0YoRiVGJSooIiIjRiUpRidGK0YlKUYpRitGJUYlKihGJEYlRidGJUYpRiUhIiIqJiIiJkYlRidGJUYl LCgqJilJInhHNiIiIiMiIiJJInlHRiZGKEYoKiYiIiZGKEYkRihGKCokKUYpRidGKEYo LCgqKCIiKCIiIkkieEc2IkYlKUkieUdGJyIiI0YlRiUqJkYqRiUpRikiIiRGJSEiIkYlRiU= expand(p-3*x*q1); LCwqKCIiIyIiIilJInhHNiJGJEYlKUkieUdGKEYkRiVGJSooIiIkRiVGJ0YlRipGJSEiIiomIiImRiVGJ0YlRiUqJiIjOkYlKUYnRixGJUYtKihGLEYlRidGJUYpRiVGLQ== expand(%-2*y*q1); LC4qKCIiJCIiIkkieEc2IkYlSSJ5R0YnRiUhIiIqJiIiJkYlRiZGJUYlKiYiIzpGJSlGJkYkRiVGKSooRiRGJUYmRiUpRigiIiNGJUYpKigiIzVGJSlGJkYxRiVGKEYlRikqJkYxRiUpRihGJEYlRik= expand(%+10*q1); LDAqKCIiJCIiIkkieEc2IkYlSSJ5R0YnRiUhIiIqJiIiJkYlRiZGJUYlKiYiIzpGJSlGJkYkRiVGKSooRiRGJUYmRiUpRigiIiNGJUYpKiZGMUYlKUYoRiRGJUYpKiYiI11GJSlGJkYxRiVGJSomIiM1RiVGMEYlRiU= expand(%+3/7*q2); LDAqKCIiJCIiIkkieEc2IkYlSSJ5R0YnRiUhIiIqJiIiJkYlRiZGJUYlKiYiIzpGJSlGJkYkRiVGKSomIyIjPyIiKEYlKUYoRiRGJUYpKiYiI11GJSlGJiIiI0YlRiUqJiIjNUYlKUYoRjdGJUYlI0YkRjJGJQ== LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.7. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> q1:=x^3*y*z-x*z^2; q2:=x*y^2*z-x*y*z; q3:=x^2*y^2-z^2; p1:=x^2*y^2*z-z^3; p2:=-x^2*y^2*z+x^2*y*z; LCYqKClJInhHNiIiIiQiIiJJInlHRiZGKEkiekdGJkYoRigqJkYlRigpRioiIiNGKCEiIg== LCYqKEkieEc2IiIiIilJInlHRiUiIiNGJkkiekdGJUYmRiYqKEYkRiZGKEYmRipGJiEiIg== LCYqJilJInhHNiIiIiMiIiIpSSJ5R0YmRidGKEYoKiQpSSJ6R0YmRidGKCEiIg== LCYqKClJInhHNiIiIiMiIiIpSSJ5R0YmRidGKEkiekdGJkYoRigqJClGKyIiJEYoISIi LCYqKClJInhHNiIiIiMiIiIpSSJ5R0YmRidGKEkiekdGJkYoISIiKihGJEYoRipGKEYrRihGKA== expand(p1-z*q3); expand(p2+x*q2); p1+p2; IiIh IiIh LCYqJClJInpHNiIiIiQiIiIhIiIqKClJInhHRiYiIiNGKEkieUdGJkYoRiVGKEYo LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.8. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> G:=[q1,q2,q3,x^2*y*z-z^3,x*z^3-x*z^2,y*z^3-z^3,x*y*z^2-x*z^2,x^2*z^2-z^4, z^5-z^4]; NyssJiooKUkieEc2IiIiJCIiIkkieUdGJ0YpSSJ6R0YnRilGKSomRiZGKSlGKyIiI0YpISIiLCYqKEYmRikpRipGLkYpRitGKUYpKihGJkYpRipGKUYrRilGLywmKiYpRiZGLkYpRjJGKUYpKiRGLUYpRi8sJiokKUYrRihGKUYvKihGNkYpRipGKUYrRilGKSwmKiZGJkYpRjpGKUYpRixGLywmKiZGKkYpRjpGKUYpRjlGLywmKihGJkYpRipGKUYtRilGKUYsRi8sJiomRjZGKUYtRilGKSokKUYrIiIlRilGLywmKiQpRisiIiZGKUYpRkRGLw== myreduce(p1,G,tdeg(x,y,z)); myreduce(p2,G,tdeg(x,y,z)); myreduce(p1-p2,G,tdeg(x,y,z)); IiIh IiIh IiIh LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.9. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">A 10.2. Algoritmus.</Text-field>
<Text-field style="Heading 2" layout="Heading 2">E 10.10. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">A 10.3. Algoritmus.</Text-field>
<Text-field style="Heading 2" layout="Heading 2">E 10.11. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field>
<Text-field style="Heading 2" layout="Heading 2">A 10.4. Algoritmus.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.12. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.13. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.14. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.15. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.16. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">A 10.5. Algoritmus.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.17. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">A 10.6. Algoritmus.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.18. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">A 10.7. Algoritmus.</Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.19. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.20. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<Text-field style="Heading 2" layout="Heading 2">E 10.21. P<Font encoding="UTF-8">\303\251lda.</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
<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>
<Text-field style="Heading 1" layout="Heading 1"><Font encoding="UTF-8">13. \305\221</Font></Text-field> LUklbXJvd0c2Iy9JK21vZHVsZW5hbWVHNiJJLFR5cGVzZXR0aW5nR0koX3N5c2xpYkdGJzYjLUkjbWlHRiQ2JVEhRicvJSdpdGFsaWNHUSV0cnVlRicvJSxtYXRodmFyaWFudEdRJ2l0YWxpY0Yn
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