feasibility of p-adic polynomials

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Feasibility of p-adic Polynomials

Davi da SilvaUniversity of Chicago

July 27, 2011

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Motivation and applications

Cryptography

Factoring rational polynomials

Number theory

Question: given a system of polynomials over Qp , whatare its roots?

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Motivation and applications

Cryptography

Factoring rational polynomials

Number theory

Question: given a system of polynomials over Qp , whatare its roots?

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

What are the p-adics?

Recall from real analysis that the real numbers R are theCauchy sequence completion of Q with respect to themetric d(x, y) = |x − y |, where | · | is the usual absolutevalue, where the normal operations on Q are naturallyextended to R.

For a fixed prime number p, we can construct a metricfrom a different absolute value function. Note that for anynonzero rational number q, we can give it a unique primefactorization, allowing negative exponents (e.g.,28/9 = 223−271).If pk appears in the factorization of q, then we define thep-adic absolute value of q to be |q|p = p−k . (E.g.,|28/9|2 = 2−2 = 1/4, |28/9|3 = 32 = 9, |28/9|5 = 5−0 = 1.)If we additionally define |0|p = 0, then the functiondp(x, y) = |x − y |p defines a metric. We define the p-adicnumbers Qp to be the completion of Q with respect to thisnew metric dp .

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

What are the p-adics?

Recall from real analysis that the real numbers R are theCauchy sequence completion of Q with respect to themetric d(x, y) = |x − y |, where | · | is the usual absolutevalue, where the normal operations on Q are naturallyextended to R.For a fixed prime number p, we can construct a metricfrom a different absolute value function. Note that for anynonzero rational number q, we can give it a unique primefactorization, allowing negative exponents (e.g.,28/9 = 223−271).

If pk appears in the factorization of q, then we define thep-adic absolute value of q to be |q|p = p−k . (E.g.,|28/9|2 = 2−2 = 1/4, |28/9|3 = 32 = 9, |28/9|5 = 5−0 = 1.)If we additionally define |0|p = 0, then the functiondp(x, y) = |x − y |p defines a metric. We define the p-adicnumbers Qp to be the completion of Q with respect to thisnew metric dp .

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

What are the p-adics?

Recall from real analysis that the real numbers R are theCauchy sequence completion of Q with respect to themetric d(x, y) = |x − y |, where | · | is the usual absolutevalue, where the normal operations on Q are naturallyextended to R.For a fixed prime number p, we can construct a metricfrom a different absolute value function. Note that for anynonzero rational number q, we can give it a unique primefactorization, allowing negative exponents (e.g.,28/9 = 223−271).If pk appears in the factorization of q, then we define thep-adic absolute value of q to be |q|p = p−k . (E.g.,|28/9|2 = 2−2 = 1/4, |28/9|3 = 32 = 9, |28/9|5 = 5−0 = 1.)

If we additionally define |0|p = 0, then the functiondp(x, y) = |x − y |p defines a metric. We define the p-adicnumbers Qp to be the completion of Q with respect to thisnew metric dp .

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

What are the p-adics?

Recall from real analysis that the real numbers R are theCauchy sequence completion of Q with respect to themetric d(x, y) = |x − y |, where | · | is the usual absolutevalue, where the normal operations on Q are naturallyextended to R.For a fixed prime number p, we can construct a metricfrom a different absolute value function. Note that for anynonzero rational number q, we can give it a unique primefactorization, allowing negative exponents (e.g.,28/9 = 223−271).If pk appears in the factorization of q, then we define thep-adic absolute value of q to be |q|p = p−k . (E.g.,|28/9|2 = 2−2 = 1/4, |28/9|3 = 32 = 9, |28/9|5 = 5−0 = 1.)If we additionally define |0|p = 0, then the functiondp(x, y) = |x − y |p defines a metric. We define the p-adicnumbers Qp to be the completion of Q with respect to thisnew metric dp .

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Basic facts

We can extend +, ·, and | · |p to Qp , turning Qp into a fieldcontaining Q.

Every p-adic number q can be expressed uniquely in theform:

∞∑i=k

aip i

where the ai are integers between 0 and p − 1. This iscalled the p-adic expansion of q.

Qp cannot be turned into an ordered field and is totallydisconnected.

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Basic facts

We can extend +, ·, and | · |p to Qp , turning Qp into a fieldcontaining Q.

Every p-adic number q can be expressed uniquely in theform:

∞∑i=k

aip i

where the ai are integers between 0 and p − 1. This iscalled the p-adic expansion of q.

Qp cannot be turned into an ordered field and is totallydisconnected.

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Basic facts

We can extend +, ·, and | · |p to Qp , turning Qp into a fieldcontaining Q.

Every p-adic number q can be expressed uniquely in theform:

∞∑i=k

aip i

where the ai are integers between 0 and p − 1. This iscalled the p-adic expansion of q.

Qp cannot be turned into an ordered field and is totallydisconnected.

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Polynomials over Qp

Systems of p-adic polynomials can be reduced to singlepolynomial equations through trickery. Over R, it is easy tosee that a system of polynomials f1, . . . , fn has a root if andonly if the following polynomial has a root:

(f1)2 + (f2)2 + · · ·+ (fn)2

Something similar, but more complicated is possible overQp

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Polynomials over Qp

Two levels of complexity: number of variables and numberof terms.

Define Fn,m to be the set of polynomials in n variables andm terms.E.g., if f(x, y) = 4 + 2x10y4 + x15y6, then f ∈ F2,3.

But we can reduce further...

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Polynomials over Qp

Two levels of complexity: number of variables and numberof terms.

Define Fn,m to be the set of polynomials in n variables andm terms.E.g., if f(x, y) = 4 + 2x10y4 + x15y6, then f ∈ F2,3.

But we can reduce further...

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Honest polynomials

Consider again f(x, y) = 4 + 2x10y4 + x15y6. We knowthat f ∈ F2,3.

Substitute z = x5y2. Then f reduces to:

g(z) = 4 + 2z2 + z3

Then we need only solve g ∈ F1,3. The set of solutions of fis then {(x, y) : x5y2 = z for some z where g(z) = 0}.

Can this be done for other polynomials?

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Honest polynomials

Consider again f(x, y) = 4 + 2x10y4 + x15y6. We knowthat f ∈ F2,3.

Substitute z = x5y2. Then f reduces to:

g(z) = 4 + 2z2 + z3

Then we need only solve g ∈ F1,3. The set of solutions of fis then {(x, y) : x5y2 = z for some z where g(z) = 0}.

Can this be done for other polynomials?

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Honest polynomials

f(x, y) = 4x0y0 + 2x10y4 + x15y6

Each term has a factor of x and a factor of y; we can thinkof the exponents on each term as a vector in Z2, namely,(0, 0), (10, 4), (15, 6) for the above three termsrespectively.If we plot these points, they form a line. We callpolynomials for which this happens dishonest , and allothers honest .Example: f(x, y) = 3 + x + 7y is honest, because itsexponent vectors (0, 0), (1, 0) and (0, 1) do not lie in a line.We can always reduce dishonest polynomials to honestones by change-of-variable methods as seen before.Thus, we can restrict study to honest polynomials. Wedenote by F ∗n,m the set of honest polynomials in nvariables and m terms.

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Newton polytope for f(x, y) = 4x0y0 + 2x10y4 + x15y6

Newton polytope for f(x, y) = 3 + x + 7y

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Simple cases

Monomialsxd1

1 · · · xdnn = 0, same as in real case (if and only if some

xi = 0 when dix , 0).

Binomialsx2 + 1 = 0?

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Simple cases

Monomialsxd1

1 · · · xdnn = 0, same as in real case (if and only if some

xi = 0 when dix , 0).

Binomialsx2 + 1 = 0?

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Hensel’s Lemma

Theorem (Hensel’s Lemma)

Let f ∈ F1,n, and suppose we have x ∈ Qp such that:

f(x) ≡ 0 mod p and

f ′(x) . 0 mod p.

Then there exists x0 ∈ Qp such that:

f(x0) = 0, and

x0 ≡ x mod p

Uses adapted Newton’s method

Can be extended to higher powers of p, multiple variables

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Hensel’s lemma: example

Consider f(x) = x2 + 1 in Q5.

f(2) = 5 ≡ 0 mod 5 f ′(2) = 4 . 0 mod 5, thus byHensel’s lemma, there exists an x0 in Q5 satisfyingx2

0 = −1.

−1 has a square root! Which proves that Q5 cannot beordered.

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Hensel’s lemma: example

Consider f(x) = x2 + 1 in Q5.

f(2) = 5 ≡ 0 mod 5 f ′(2) = 4 . 0 mod 5, thus byHensel’s lemma, there exists an x0 in Q5 satisfyingx2

0 = −1.

−1 has a square root! Which proves that Q5 cannot beordered.

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Hensel’s lemma: example

Consider f(x) = x2 + 1 in Q5.

f(2) = 5 ≡ 0 mod 5 f ′(2) = 4 . 0 mod 5, thus byHensel’s lemma, there exists an x0 in Q5 satisfyingx2

0 = −1.

−1 has a square root! Which proves that Q5 cannot beordered.

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

More complicated polynomials

We can apply a version of Hensel’s lemma more generally.

Theorem (Birch and McCann)

Given a polynomial f in any number of variables over [Q]p ,there exists an integer D(f) such that if for some x we have

|f(x)|p < |D(f)|p

then we can refine x to a true root of f . Moreover, we cancalculate D(f).

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Birch and McCann

Good news: we can, in finite time, check if a polynomialhas a root. Just brute force check for:

f(x) ≡ 0 mod pR

where pR > |D(f)|−1p .

Bad news: D(f) is resource-intensive to calculate. If n isthe number of variables and d the degree, then

L(D(f)) < (2ndL(f))(2d)4nn!

taking f(x) = x7 + 4, we get:

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Birch and McCann

Good news: we can, in finite time, check if a polynomialhas a root. Just brute force check for:

f(x) ≡ 0 mod pR

where pR > |D(f)|−1p .

Bad news: D(f) is resource-intensive to calculate. If n isthe number of variables and d the degree, then

L(D(f)) < (2ndL(f))(2d)4nn!

taking f(x) = x7 + 4, we get:

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

L(D(f)) <19333264281334959478690053515795664423387946764381450654542262551041804628897211821857886513577419545858439735358669698984030651760105412130716409572405122442658961797539822338952893590140699079166598663682148017168884644056031637925155326893646940162312704215189827992536737192946871426590561959658753898205091640503148428908169392425185932175576810 33764453087920923956369810667371288974205526385135367143464561906167502970726504712524486476057166330728844298342919517364978030022251473663 88190637560540189965784127958446926663503878599874585771220380465820307761161992648760557018169228488048159680497348475599240382107647372869 53127736909063180888219962845900888170959843155649304800498477523622912423143837913284235360086572243436971731852708267608608416224987193888 86774621995452049705556011357005923477639435226450730541308569266773792530090205227886457244539696443077971220811050217647331617753931550473 74427006169667107504054190872534784859366597113730703173153900799873993824882162922569391197219345384430610117124584911245229330939654415004 27493506357399752980835300969261554659869198880886332653495415819809788011918846230441297647208396630888129631926621760950435205570044116659 20956163667630775508269905675090697561567681462111579966111671944332247877574871219345599338919610932098495256107502874357513111580297287287 50174173914016470595640676113325746643660474886284383532804616262475554950066460741775965193149538964142854642578728529899818673738874796709 84452149905653535579843058460191492084277518160815362298328773416573977668178575234627495103485837349022565685845524720865993770884625119910 74058143334259521325704752535688285563533465000511554537046680670026297241759069891554646175191972671878828350001258276425170821998791185113 98789237776423943562176546998152333847554013767178346626233841492723131320177769255664094994575716401817267792819131529949168032944957511541 56175907291869761367905077005195577050948345742927074448723711771828883349446731832134961647515366095987698998487383009238538840803314159656 85818253966971057431356935012850657615484161750002284975870971291644429827033566051394313338149330768459510942242979383995013895360809123904 67030093385641168029565433692675409810864099677033241527546321996935643789412419321547220236549888490281529589008816784610245865704248022588 74353313431293633525604213832630172566985850850583549591693666390699675117327358126484805714690040674385421597109612903329678594555493335001 13049595453542761946615692925677513324222034329747143058478730145092300399191510424525279150769097652553148263820279250828459068579674941950 89019114155246502840101536623337590448655892980256477817473507729841133169586496626760414000330198843819694296841079735426025475416495720510 20527182449545104967887793621298761577152281339945065120590008509440016474155910297918835391398923683986493677650251936004575387818689363688 67320772507279373085174827443461202863330918824543787162408791951491580499366898312839571513071389651061533372015688742785032852250207936198 71755656566646604415526757573110530569081255717807547284110849262898232214316009301522994672051315282706792270482251971164290477935589863473 17384406738558550742891818181102975857077200680978937578028801558796949106504454062855087485450284643979346120040531744059090084373803645025 99294358330576408001403494142659939565667792471621353269079509046486580675580027066187887338364379957201763353854223804520124778777814477912 89769688735392266233435679306193805989831174177447130405579159013727180175028044065660381812015102977968112688031951783579489534741050738872 04659640689718003011227780976729150792189538704571126176974089414249355746466960506239155665268750324608746540891226250399065114160736002020 25608454573224922982557754757544765192855251007607562958599094844144457882727651119842761188037971152914466738633927204203067903374584227372 20783736556929346414352406458395598020341650746124219231944751776047053466229911717153016824182022205447176953710557428643965545073094155945 81387294166285727569456107001116284178512189774612521714512784613630304580007333344148260808345326229548535122591715680334219522227727531010 005639255540923363761797

Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

2776870538765384880850236686657758479110571426 66881598470090481050677593054802693440880997883314349466303470127750098598203607316316083293247786796357682195903001518653362172094100831501 50416009282139073548016549939169986120679046776519260597766334727638224731846568268030996254461300173875140624480328425187623808170263997372 09935791122989075431392880199666651535427122964093119609902615352133614233656270034709180593498687112465756128403365728728088698224866268256 24181135829323243249064092498228495196226133892068987681627268241598271817586210591019592386616553200363801575976790385248518829547445513064 65006044123578635375193925132201757084418380725264906517259321188332737907429741134157430677568646547474957039660069297603195720684793658185 36493982865702214229163008591658960409377322165874812642069224620035832582688210540150690415468773245456764938676331366478753370880541418571 60728201571628066149569052234686498837090223205180963259589979906825140347551407112314077091088161735346213874016140702048096491026734136636 38669433192190102990264743310463560589034691939310354520736830781238972717838427929257231997877025120194234188558328943090473231807066753637 36102993815162707633682796595609322246126939171375139546815844018565853497776479417604199556663028415350911614928040646055840625465301676548 19190121182407850871241184931835035387357799215528249863011999142482108202019656975633237627797072174290345530628310814027176575088405456921 52731097475133140929041440351130289721690769577748145199946443923907196348162257122881493750403460810540250190946966961141002201643558592392 36324093475381166774036651484629803823758649920824599814400993110529426132346008100875793157790041247049511101603725632151413699501152621197 78643274273058599220119548242392150117347612302406868919379272227674006992611976921660797046531766253590665143278669193055249065058801541289 68498195626718805917327562760458420798161573330682366427638017156302252302647391317157021711875489003439329522982168415388727510972464314106 58094748733121608969820541308612845596005255481294794025585762735245995076734671914944940782977151831293164311904657637368573917298425732961 35245164443319446361325443371197717209035665494159644812517338391650694363166765395897112736170705883937184291555006766938068339666233574486 30205872845533934074288451323207581755376964514594423488722380559995188787390474721746810282763613588158910419939546071908233655282222784783 99497978605777529043928817383709908382772627814683993733822318675567493782122179196080800235862012927332241392443147482116066746391805873932 86960726749568777492846330593622157178898576361544884766607137362685752157724008796460684722451833909727277552705297113731137200940590141818 42525931879736693053120699387826877059334724288418068954985682033771848945663567731633459264955361248627327112687776834054166202827465667420 82437364249844893356346131436719409558229760945808719842211717096774188911248439774078955796487942868245961948022315558728356719911468612508 24682092046587951074595824064773850366572293083696097335455083460968782433574676956473102148784275911787807732931690814649204490278476720703 63445298265370606430991650948382729272262308059070531019503783684192631413823533740904173560456662797153891800967176467846248719614240267779 03179672553711612771133136242150163099912570890134891245386064927853821421381625317708356140777507446398022656364172531680424542429031620648 39986605168659756323769134513089659030935511921455124893119649282909503701473139325194814013003001410709377319471053338338976687410348966754 64866762952070456631552140007979671251505785990845978114413181139801999119548323167770083127402823062823434612036160305993803379050870402972 50561473920523993435915883541936349660249791137050887392983267942364542868366719084556611626896479348569979476088133185774631619278338842922 15098340060258841070100937832566984972136339296726959189797228310469194537869857114784177040762903353812250606190205366894032100379717349104 065261430937893744944195

Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

4551942021091893315046582487397778114102740801 28700751841565157654200943462776118894702386784436263382495897872086352685518244394188204540141287071531832668108835353447180803864257950835 48005776165783640956649540026189506148664220809117385565914304783230408921581750897046105875263423274341134280008680498272389043329299124763 24451327926634862713546618548842768306162952999188892337086651320827499491078284960131973849224931991173322743277977413512404191667984602906 61530749778997727378355722151719453593733125795624104085937521483652637986542134517133910075032188637732275234029805130660935874662855959492 33014180083332925323139860076856372151725787993249860699266079876018185333047962320439733280363688402448393283160624104349053309994029946516 54178287402665209806868234469672467349641288583129517323667633829295830133956700670542877366085497270431101337962307541619535454942878984823 02463145141143531507751671743284185325543593893285243512369585015080382012070637341894811249663271899917592170940381209119342678566768003503 52066435055945234507989849516732773401853417376599513587950602464377316502241502526713020475266318316765586193597329207863536907595606123799 22626959416671018802482341700044254117941778904560559194796871472688079622449766218636029191906195577932407980135466833969496624040124843449 66443499830356212459957202640850721716755631215785586684451047534075302472926216209467798290985426363226746020920881874576325567137568696360 34028639767482731406861997703694615581248394513975987414813208649886376642751409721311959318990898567051595729288753041692997497726786776787 76665780006044319102698955076222546580327692186015498340022794306089295146916184453460928039963292547510599979832884205964074785590540141837 77836551597971682649671142452041332426931044731074455699045923189732788038343759836539876312800296043537050973083174980001672055931722611853 44018394811324040253410849641775045489477869470032273993021926947339516165601055844482426420971617585219502951952603180185843459857422239317 67159184565886775198196981423241070130124535265068367126377560936071454075912707130237858511272581698736398278993723959122146129628657983577 52084590840267080593912712628517403124548944368492240145363556831270927009422742494596662504954799596383541037946452743275979447367026249361 28917983696919013015172148388320805823122694753507144892295718035686455710685557304016443177938347849178261208604558636478869398460620307716 48915329742625249703356039446321752358052142027709947588836308488041953785252109654884535122050706836417167686568383080726299669648450112620 76131766203717688485532333782738137981557149670005724676730624489831382372854161296432137383203602302892795329353462748398745976357965484916 51404081455101690579687397543134292577873698987784497712149295625539664401420295329861549724996214850019901641793735244392199477734135012413 12264291451808295244415295583472659497027057032923754179690121257389137919834495511336751551833436225889582099931180735639547771819072089269 20533110915059544524413186386298223242744635061015036890199290185910762847816320411705439888346906101020628960920759310335161319585780675532 63493547875825186150870900809166841340364629138463998603696160539952771086877493610527761447194233697953316826998399779374766758801826205367 15894056219071623371902273249273422519576921444786304257945976400366276543511217996939592481390814910120220576876612743453849390322954894769 94063505960092908958240765109231818355475791768239586385427994173065015888309147483415772282335022074415179733036699579983472842447980716356 82233221723166501345930030235982785069027620004393167729748230192189159041501062569385740085497539124231737227329432947011927354180251786879 40642001258393812674614583888067926968479532455273656803624385418276391695775313324459085996358921010174431727332982945463813543156612632508 44893719227486365094093642458171402601772056025320020646405486341464178953907671884804763187705163750951698671390248312022915784478142101053 631622424509275875982395

Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

4981499329734625753956056408466254565855807880 41236075423753961168443589136840801742801470191482723941232108606027875928309054801248486292561098139657284944466082012135932641240835876708 21847116974565255868318426111019935477317159404608566704806206919961326009310526284468778703179786543021247185894066821715033013600196181513 34589891362021812423757538066728984958968821260392104779958254086791819887552038268612696911755419194394725935929625976261687472889791141638 88618372810430564394465416399313113723817407069473896125928508391162892983760339622802780125162672489764200362758344685630093770055060741060 76200124611831131989334796992424459100791986304208799308455904084930426716506514025379996483030608324172623555255326909648198100374876418304 53525212982282605386684374851352953969849377813012192519704844346836839019874114167929509509173480440815541564923141182359372409118644544732 08430051600214687327211635980633642115041442772289443888200623107311315581369877615686034360876930273543338201753898290985560860760981506111 35250624379456608932374119908468027118249760328896518566378350964899588555455622494975399278363613444740752987111006979598141751516216118734 90388432254708560793695924527664016065121114829331411291916724114373849088785230456241835849268578883769845115694129636217642725977720850251 69123090167106676031446987174769964995482693429113226858674031607792002006315387695304307696631812330404859286569857224365099301333167812240 40759987870899428185110097776237529656756014650759295275864327992258952907184121892327134509956170592235148257374494966000030800513040305842 14173022419621576268327577671083859624250967011276989845651269428389524282227006426878484585132636753334520888930866731162170669630885288947 31313743030597883186013725455815243935009271150909247869809914266995996370276678632156613075326973827465731371851496924476827601783224541117 66036522508354155875035284435866970527066677295998493469365702099032391638812172848855092635565413574159829951572539703092215072741841807056 83443353286197332510725728160744160460638703509987475761367892513496203491930529582856966091371776210150946261256284258255556802208436555942 70932064178136924422151462265247596221887688786934335277256448647093735935098890771278152305206115997893526683576513273378475013627418690457 29356618809627902694845553635404120682557871613379191950192062151085313244764782243887823486962712664063775831018884755806580476266182826472 52837445857013102586538020474661207237536832756085714085839436448821114682589771425886376600047632568685627883087087466633200688798182340570 11286107158670808446396855993589065557253656751805284968244597460544707465794756938575979957710217670938072888485422191914636416862806942091 02866561674907421629221319682380204407255043281483216875109658779314669013670720793670410130459850164000150045389752692140585220900441045644 72494833299344337960050740045253162906848409733153836550842484630861699178481261506414525413296217795933518952004771244780403441977357143147 29312376907575919238359999977359138890037099304191020575626243834387680086076433038737442785063322610345425510972514528801944016234737388323 03158748111943608713152892710277164432996692087887157730425122616305799493817711567467416785996468096229672026992938040310717063437525442653 72953155760420546534692838256935549579538230398902783409910491470986685242075515310118427054165809046196614908485780377719880641848311091668 97701118235658149944635093092332803724775083184084301445126516092158733586199260108567459723387788949465123595538222677165433011388971469578 67328111011097297865805682155360267648108361459470170122424562794620057269452367791169552797256332057936803753229342159625070574017613584033 10629066668913803922288445699728224590103843221410009615643216110710425533128362653139838192214806564710071930544384271195308540742333806169 88257217977276879282463800229823305242534899046152077053106503605603818069503805249506876655159191076166515836255389617798602831995998968095

Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

9258569189421579248896856071334730561897959614452315812016458345276275 81922545570762265765142404956659460120525724175822876964790473426285784750016386128182627818451574714826448540531212380982773234558981478939 93611678012503076795256555547151898909084378558569378909970250573621311410109219719584140746947407093180153154716007934873672193053764621482 00897243913244901781078278398652625752955135309872736259599779493739151017096211807581786340102542674303493576417353270321262104393045939175 98523112084949227064704228210360027701406668877221071725091445594753868850779300591592625223568222936047134259113232323617758192275451621117 09289909623752210632799037132739096875874069972443909874473870474720190717154815040766507255570456978561517958891352342515792731266291633373 11318233449830954629363630073432096619067263458779478971063166485014501527704840359327341209375579332711362022023030290270919346971830858002 72644366250837812984322070884063394302585507671739072855424951902039651007786336248236776985785619828577367009966060476590825523642025151934 99700322494360898318051424345626433698559802756706940184090221209208821251385397748099806369394361868251837656564999550021086755706652361564 48059462993588729652526810462808896187929738363748082122394045159156056787029179457110904283865468305849945233218958064333605993464701290217 61995956933876920492955924599523614231169347935751846689553364020206257116469023990635871018497682049117591825215915280537815915871962893198 09624869901405190609515122221522051983632258927702179604308583851266716624532619480842013070311806548223666815029094102062730669036550259386 36588840269815018617823749949248526107709652352305938301317471075046765727228114933230059567795880662731346520428145543992531173899770920088 98294409921106875279355824005730703602559413991549360909538408983709008343156375221804788803563756996449738681623729312217421535902950467261 70573920925479711468844737785697470385022797971903928363870448062432360598895372923229171299325815273972290059598281501805853176974464464020 48601720999393889116545923819030599487015872491745766250177324315920019350507175299173524917421313603698106554426707217938259727506627385348 74693805092179637235538621492198525745781505547861523562697055568075427582514271379868665747454582623807900587044429890165506827155306306571 26442126852536918861711970955212244656195331650918762467970634421004978109890282824383408385485490725447498125742637978395466623516461462809 40236636037065830824288444221110353581035591737701571278344255798989136694333324288723276763636471983316000037700956435118678397573774769878 38710098460056366794868741099979072708414574668288814078042149042375107397920285310853739748325041833382776748347936616663172705910102576712 02163808754750742950349377072312032116483095376348082029270415804529997671123864535245084097229358710254565652838504860969861755888459168529 47325794468976750993515050263719082118436428044325187417309904508225917209991932706811056011820661743340005563203390572447494043978246308658 44739272103904884138159459414067067567417653547992459793060152975217286352846154667776280575820666901510805471839253416474258687312371960862 31965135309055205239410258677533852630061858198655633747040416868618216824617911767714089655710718050760519806128710542068921241876884472869 47584448647725254674685218301099840235496656340540023729522625245446108602985677616600341949288841987509029539428042321736958370474486865212 93181047875730375260033772548445553641427436515624125926162503970885023049466752679447682703639607172365967127147323092638568128000247203007 77855813127964850835727360398445595417592620165863724281323724103426788829267366454417373343825993756449773116118717968933558628939365169806 67096158071350687756414669769020894906537242640356478771890299850051597815777044556384498552806649943510425361801462614177381401881391551683 11013783891836223136452207241550789338750777834456437545017112170467670344584923456616307202138982006731832473778779286103310591826580086935 5640438299135807480491138669759012750360422354086113699113146691464137

Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

7199188373246741206605662801845319362060219134350275144916498005561185 43541739896312740684473694207491244532354239139224878717686711994795295538620550159360799713028854768438041523993496680243159594865969623349 50988598328403754755588664392226586199880895816930570838659979022892824282948266930427067312266731837490218714684892871738100493261649529839 51712607406260344015233530898132454884509415904964384636391430723705171904567686533547695258057556529762684247408869236632016577754326954093 22077254519118582446088502581801973248700497474897113500403224067469788382903329116638642694709861429945910962893254169827026500193379290507 26557674381422315050042768285607870105862709195298402557541974236878833059454155099652315999917772543786265038710239359963379244033365151731 91080813593228366264511906675401380280960257653902094980092387565565263728142063493966638550293417192037419090382748878522357286489270695126 70335026676122538051388340227378644479231486170161957921977119753627994474494157361045369193981305946055505242046508174899347373502807919216 22513310736051536711217504026618753088842465392372914753149368795556569451139108131566649581159196400487942775849692765994637669846977935071 26440452005819032654532006039350369187305290551993056086280221111480578603389475304124976992255439235798922936179059319476653541800250591617 05574624434194278897094655822621168510632343469550157649759721567480052072884378399426521794656412568689321087294219610252736870006284081581 45302009617848980051828143538412475172290779392713662452989662863973250564015402000593209088756461030608775460159145662775675729930496629315 52367855473648273548072027374035668875414381888567668698723157412211588685050449983283316889070625920757243214457091817699287375536927414745 10622918328579913707603169889291381596467176444556132160228833957212116278883362246691879734019313491117063081318900131028527973178631567546 41243747559866076594311536666777575362424168783399145616749570569773570238502333563561904336211374115633317502339088378404419179762591439327 40608182314524784846851592097557031319557726486913052223650516572231323971068837470718402236792978382026600980199675909467363203546454674990 28692108531049192314727981562381670557729582809365438004019190127080010278755850894558206116429818462011053557073212761035286647747807963540 13969200485292725340971365567520652320721539480220334830756841935850717725426345146956701018294688315450061189086730205218119081036790348152 68632476756934347822164472302148192720110129891050174456874579543083349015466789889714268406717359828844402096033897759983872158916658254816 81435615061892226610259644793803825361050991004053214411642520929119175273199787639340087920590289639360277067412915200195263965247512822498 69306364645459357789945744637611916870044629212670811289393805318805508887091657458828605013087610309080738617604412507010602465972099204920 82249508841077585044805520961583876393090752171709763669211886686345725669686424215682855430282139431917902045975913113849145474067646281070 19036821740898095616985949717481593060425861954672758670209784879024047835317856209812906951200524451707221275990672778236299023304965806815 02679979960011315776055150816108005662255732910231857894265282831875618210622344783311114700198503061053952743901132834902116905644225345550 79947339854197547673448416415407396766793101965172964088559836683009028319978464062783091149653770204702544317112890715322760473604803001373 03390563661330180685323667019939354757074429076590729473947469164198298772214025516731598169016766669903755887525430684490625459370870357507 85764467540867773396685482445664198222258230437996892226120590909525907210949705964050091695973580249262485118940967059521205606504799313985 07334702948586702474004522632580080870480414456295216639460557886479570130888647292924606264538178213031040820515644282770463889992688266035 19854724652896561502296155394739684288750993074146542679838111833008065931683798331665646581179999108917791761114099001769238562209575112929 2042257949658219115499480409009012584403263847900996186955078922518132

Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

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Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

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Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

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Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

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Really big!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

More sophisticated results

We can do better, though:

Theorem (Avendano, Ibrahim, Rojas, Rusek)

For a fixed prime p, finding a root to a function in F1,3 is NP.Furthermore, allowing p to vary, finding roots for almost allpolynomials in one variable with integer coefficients is NP, as itis for

⋃n F

∗n,n+1.

NC ⊆ P ⊆ NP ⊆ EXPTIME

For f(x) = 4 + x7, taking p = 2, finding a root wouldrequire checking for a root to an associated polynomialover Z/32Z. Much better!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

More sophisticated results

We can do better, though:

Theorem (Avendano, Ibrahim, Rojas, Rusek)

For a fixed prime p, finding a root to a function in F1,3 is NP.Furthermore, allowing p to vary, finding roots for almost allpolynomials in one variable with integer coefficients is NP, as itis for

⋃n F

∗n,n+1.

NC ⊆ P ⊆ NP ⊆ EXPTIME

For f(x) = 4 + x7, taking p = 2, finding a root wouldrequire checking for a root to an associated polynomialover Z/32Z. Much better!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

More sophisticated results

We can do better, though:

Theorem (Avendano, Ibrahim, Rojas, Rusek)

For a fixed prime p, finding a root to a function in F1,3 is NP.Furthermore, allowing p to vary, finding roots for almost allpolynomials in one variable with integer coefficients is NP, as itis for

⋃n F

∗n,n+1.

NC ⊆ P ⊆ NP ⊆ EXPTIME

For f(x) = 4 + x7, taking p = 2, finding a root wouldrequire checking for a root to an associated polynomialover Z/32Z. Much better!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

More sophisticated results

We can do better, though:

Theorem (Avendano, Ibrahim, Rojas, Rusek)

For a fixed prime p, finding a root to a function in F1,3 is NP.Furthermore, allowing p to vary, finding roots for almost allpolynomials in one variable with integer coefficients is NP, as itis for

⋃n F

∗n,n+1.

NC ⊆ P ⊆ NP ⊆ EXPTIME

For f(x) = 4 + x7, taking p = 2, finding a root wouldrequire checking for a root to an associated polynomialover Z/32Z. Much better!

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Neat result

Define the p-adic Newton polytope of a polynomialf(x) = a0xd0 + · · ·+ anxdn to be the convex hull of{(di ,− logp |ai |p) : i = 0, . . . , n}.It can be shown that if a lower edge of the p-adic Newtonpolytope has an inner normal vector of the form (1, k) andhorizontal length m, then f has exactly m roots with p-adicabsolute value pk in Cp .

Consider f(x) = 1875 − (24875x)/4 + (24125x2)/4 −(9605x3)/4 + (1783x4)/4 − 37x5 + x6. Then the 2-adic Newtonpolytope tells us that there is one root with absolute value 1/4,two with absolute value 2, and ohe with absolute value 1. Theroots of f are 20, 1/2,3/2, and 5 (with multiplicity three).

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Neat result

Define the p-adic Newton polytope of a polynomialf(x) = a0xd0 + · · ·+ anxdn to be the convex hull of{(di ,− logp |ai |p) : i = 0, . . . , n}.It can be shown that if a lower edge of the p-adic Newtonpolytope has an inner normal vector of the form (1, k) andhorizontal length m, then f has exactly m roots with p-adicabsolute value pk in Cp .Consider f(x) = 1875 − (24875x)/4 + (24125x2)/4 −(9605x3)/4 + (1783x4)/4 − 37x5 + x6. Then the 2-adic Newtonpolytope tells us that there is one root with absolute value 1/4,two with absolute value 2, and ohe with absolute value 1. Theroots of f are 20, 1/2,3/2, and 5 (with multiplicity three).

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

Acknowledgements

I would like to thank Dr. Rojas for his knowledge and support.

Feasibility ofp-adic

Polynomials

Davi da SilvaUniversity of

Chicago

References

Martın Avendano, Ashraf Ibrahim, J. Maurice Rojas, andKorben Rusek. Faster p-adic feasibility for certainmultivariate sparse polynomials. Preprint, 2011.

B.J. Birch and K McCann. A Criterion for the p-adicSolubility of Diophantine Equations. The Quarterly Journalof Mathematics, Oxford Series, Vol. 18. 1967.

Marvin J. Greenberg. Strictly Local Solutions ofDiophantine Equations. Pacific Journal of Mathematics,Vol. 51, No. 1. 1974.