On the automatizability of Polynomial Calculus

8/2/2019 On the automatizability of Polynomial Calculus

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5/12

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zlff ab #z ' q,$Ekylpeymq{ @mq eff l Wu_ zb kmd{libq e ~m{{laf| ~eqylef `rdkylmd El 8 w>& 5}{ Wq_ zclkc

{bfbky{ mdb m` ycb klqkrly eomdj Kl5 Klq$ Mdf| `mq jeyb{ ld ycb {bfbkybi klqkrly zb bd{rqb ym~qm~ejeyb kmqqbkyf| ycqmrjc ycb klqkrly$ Ycb mycbq mdb{ eqb qbflbpbi `qmo ycb kmo~ryeylmd$

Jbdbqeymq{ @mq h Wd_ Jh 8 w>& 5}{ w>& 5}u jbdbqeyb{ e kmfrod pbkymq m` E$ Ycr{wJ5 Jd} ibstdb e u d oeyqls$ @mq eff l Wu_ ycb lyc qmz m` {rkc oeyqls l{ `bi ym Kl5 Klq e{ld~ry$

Jbdbqeymq edi Ekylpeymq{ zlyc q/{rqhbkylply| edi ~eqeobybq {$ E{ ld W1_ #{bb ycbqb `mq ed bs/ybd{lpb il{kr{{lmd eamry ycl{ qburlqbobdy, zb dbbi amyc ekylpeymq{ edi jbdbqeymq{ ym ab q/{rqhbkylpb`rdkylmd{ md imoeld w>& 5}{& `mq e {rlyeafb {$ Cbqb l{ e kmd{yqrkylmd #{bb ef{m W1_& ~qmm` Fbooe;$5 #ll,,& ycey effmz{ r{ ym ibstdb {rqhbkylpb `rdkylmd{$ Kmd{libq e jbdbqef aldeq| {rqhbkylpb `rdkylmdj 8 w>& 5}f [ edi e ibkmildj `rdkylmd I 8 w>& 5}f w>& 5}f m` e kmib zlyc il{yedkb 1f #kmd/{yqrkylmd{ bsl{y W11_ {rkc ycey l{ e rdlpbq{ef kmd{yedy,$ Ycbd j

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;$5 ^qm~m{lylmdef BdkmildjZb dmz ib{kqlab ycb ~mf|dmolef{ zclkc bdkmib ycb dbjeylmd m` ycb ~qldkl~fb il{kr{{bi {m èq$ Ib/

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2


6/12

q l{ e ~eqeobybq$

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sh #,|l#,xhl&g ? > l`` ycb l

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klqkrly l{ xbqm$

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l @l& zcbqb K l{ e omdmymdb

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hl&g ? > l{ ld ycb ~qldkl~fb$ Ycbd `mq ed| edi l ycbqb l{ ey

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0


7/12

|l#, ? > l`l{ dmy ld ycb imoeld m`El$ Ld ymyef zb cepb d1M#{,' vKv1M#{,, bureylmd{$ Dmylkb ycey

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5$

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5$ G l{ e fldbeq m~bqeymq$

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;$ L`libj#y, 6 q o ycbd G#sy, ? G#sG#y,, cmfi{ `mq ed| peqleafb s$

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Ym ~qmpb #;, zb dbbi ycb `mffmzldj qb{rfy{ eamry libef{8

:


8/12

Fbooe ;$ @mq ed| ~mf|dmolef ~ edi libef L jbdbqeybi a| u5& u1& $ $ $ & uo& eff peqleafb{ e~~beqldjld QL#~, ef{m mkkrq ld ~& u5& u1& $ $ $ & uo$

^qmm`$ Fby s ab e peqleafb mkkrqqldj ld QL#~, edi dmy ld ~ dmq ld ed| jbdbqeymq ul$ A| ibstdlylmd~ QL#~, ?

l clul `mq {mob ~mf|dmolef{ cl ycr{ a| {byyldj s ym > zb mayeld ~ QL#~, s?>?

l#cl s?>,ul zcbqb QL#~, s?> l{ {yqlkyf| {oeffbq yced QL#~,$ Ycl{ kmdyqeilky{ ycb èky QL#~, l{

ycb oldloro ekkmqildj ym 6^$

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? l$ Ycr{ a| Fbooe ; ed| ybqo ld QL#o1,#o1, imb{ dmy kmdyeld omqb m`{rkc peqleafb{ yced o1 ly{bf` kmdyeld{$ [rkc l

/ldibsbi peqleafb{ eqb ycbd kmdyeldbi ld ycb rdlmd m`

ycb peqleafb mò5 edi o1$ Ycr{ l L#o5o1, ary ycl{ kmdyqeilky{ ycb e{{ro~ylmd$

Dbsy fbooe l{ ycb cbeqy m` ycb eqjrobdy8 ly {cmz{ cmz e {oeff ldibs/ibjqbb ibqlpeylmd ce{ fmkef

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Fbooe 9$ Fby y ab e omdmolef m` ldibs/ibjqbb fb{{ yced qo edi L#y,

L zlyc

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v o$ Ycbd

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Zb dmz kmd{libq H ycb {by ld ldibsb{ h Wd_ {rkc ycey y kmdyeld{ fb{{ yced q peqleafb{ eomdjsh5& $ $ $ & sh{ $ Ycr{ vHv d o$

Dmylkb ycey abker{b m` q/{rqhbkylply| m` jbdbqeymq{ zb cepb ycey `mq ed| h

H edi ed| pbkymq

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hlg ? > `mq ed| h H& l L& g Wu_$ Mycbq bureylmd{ eqb fb`y rdymrkcbi$

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Zb {by xhl>g ym > zcbdh H edi 5 mycbqzl{b$ Ycbd zb ~qm~ejeyb pefrb{ eomdj ycb klqkrly bureylmd{

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Fbooe 2$ L` ycb ldibs/ibjqbb m` e omdmolef y l{ fb{{ yced qo ycbd G#sy, ? G#sG#y,,

^qmm`$ Kmd{libq e omdmolef y m` ldibs/ibjqbb fb{{ yced qo$ Zb zlff ~qmpb ycey amyc G#sy, ediG#sG#y,, eqb buref ym QL#sy,#sG#y,,$ Kmd{libq ycb `mffmzldj kceld m` bureylmd{$

3


9/12

G#sy, ? QL#sy,#sy, #0,

? QL#sy,#sQL#sy,#y,, #:,

? QL#sy,#sQL#y,#y,, #3,

? QL#sy,#sG#y,, #


10/12

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On the automatizability of Polynomial Calculus