Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

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University of Groningen

Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

Baker, Matthew; Lorscheid, Oliver

Published in: Journal of algebra DOI:

10.1016/j.jalgebra.2020.10.024

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Baker, M., & Lorscheid, O. (2021). Descartes' rule of signs, Newton polygons, and polynomials over hyperfields. Journal of algebra, 569, 416-441. https://doi.org/10.1016/j.jalgebra.2020.10.024

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Contents lists available atScienceDirect

Journal

of

Algebra

www.elsevier.com/locate/jalgebra

Descartes’

rule

of

signs,

Newton

polygons,

and

polynomials

over

hyperﬁelds

✩

Matthew Bakera,∗_{, Oliver Lorscheid}b

a _School_of_Mathematics,_Georgia_Institute_of_Technology,_Atlanta,_USA b _Instituto_Nacional_de_Matemática_Pura_e_Aplicada,_Rio_de_Janeiro,_Brazil

a r t i c l e i n f o a bs t r a c t

Article history:

Received11June2019

Availableonline4November2020 CommunicatedbyLouisRowen

Keywords: Hyperﬁelds Polynomials Orderedblueprints Newtonpolygons Valuations Tropicalalgebra

In this note, wedevelop a theoryof multiplicities of roots for polynomials over hyperﬁelds anduse this to provide a uniﬁed andconceptualproofofbothDescartes’ruleofsigns andNewton’s“polygonrule”.

Introduction

Given areal polynomialp∈ R[T ],Descartes’ rule of signs provides anupperbound

for the number of positive (resp. negative) real roots of p in terms of the signs of the

coeﬃcients ofp.Speciﬁcally,the numberof positive realrootsof p (counting

multiplic-✩ _{Acknowledgments. We}_thank_Philipp_Jell_for_pointing_out_Remark_1.10_,_Ziqi_Liu_for_sharing_with_us_the

examplefromAppendixA.2,andStéphaneGaubertandananonymousrefereeforseveralremarks,including explanationsontherelationsofcertainhyperﬁeldswithsupertropicalandsymmetrizedsemirings.Wealso thankTrevorGunn,JaiungJun,YoavLen,SamPayneandThorWittichfortheircommentsonanearly draftversionofthismanuscript.TheﬁrstauthorwassupportedbyNSFGrantDMS-1529573andaSimons FoundationFellowship(Grant708029).

* Correspondingauthor.

E-mailaddresses:mbaker@math.gatech.edu(M. Baker),oliver@impa.br(O. Lorscheid).

https://doi.org/10.1016/j.jalgebra.2020.10.024

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ities)is bounded aboveby the numberof sign changes in the coeﬃcients of p(T ), and

the number of negative roots is bounded above by the number of sign changes inthe

coeﬃcientsofp(−T ).

Anotherclassical “rule”, which isless well knownto mathematicians in generalbut

isusedquiteofteninnumbertheory,isNewton’s polygonrule.Thisrule concerns

poly-nomials over ﬁelds equipped with a valuation, which is a function v : K → R∪ {∞}

satisfying

• v(a)=∞ ifandonlyifa= 0

• v(ab)= v(a)+ v(b)

• v(a+ b) min{v(a),v(b)},withequality ifv(a)= v(b)

foralla,b∈ K.

Anexampleisthep-adicvaluationvp onQ,wherep isaprimenumber,givenbythe

formulavp(s/t)= ordp(s)− ordp(t),whereordp(n) isthemaximumpowerofp dividing

anonzerointegern.

Another example is the T -adic valuation vT on k(T ), for any ﬁeld k, given by

vT(f /g) = ordT(f )− ordT(g), where ordT(f ) is the maximum power of T dividing

anonzeropolynomialf ∈ k[T ].

Given aﬁeldK,avaluation v onK, andapolynomialp∈ K[T ],Newton’s polygon

ruleprovidesanupperbound forthenumberofroots(againcountingmultiplicities)of

p havinga given valuation s in terms of the valuations of the coeﬃcientsof p. In this

case, the rule is morecomplicated than inthe case K = R; theupper bound νs(p) is

thelengthofthe projectiontothe x-axisoftheunique segmentoftheNewton polygon

ofp havingslope−s (ifsuchasegmentexists),or zero(ifnosuchsegmentexists).(See

Deﬁnition1.15foradeﬁnitionoftheNewtonpolygonofp.)Ifp splitsintolinearfactors

overK,theupperboundprovidedbyNewton’sruleisinfactanequality.

Thepurpose of thisnote is to provide aconceptualuniﬁcation of these two

similar-looking,and yetseeminglyrather diﬀerent,upperboundsviathetheoryofhyperﬁelds.

Hyperﬁeldsareageneralizationofﬁeldswhereadditionisallowedtobemulti-valued.

Given a hyperﬁeld F and apolynomial p overF (by which we simply mean a formal

expression of the form n_i=0ciTi with ci ∈ F ), we will deﬁne what it means for an

element a∈ F tobe aroot of p,andmoregenerallywe willdeﬁne themultiplicity ofa

asarootofp.Wedenotethismultiplicity bymulta(p).

Inthecaseofthesignhyperﬁeld S,wewillﬁndthatthemultiplicity of1 as arootof

p∈ S[T ] isjustthe numberof signchangesinthe coeﬃcientsof p. Andinthe caseof thetropical hyperﬁeld T , wewill see thatthemultiplicity of s asaroot ofp∈ T[T ] is

preciselyνs(p).

Moreover, if K is a field (considered as a hyperfield), f : K → F is a hyperfield

homomorphism,andp∈ K[T ] isapolynomial,ourdeﬁnitionofmultiplicitieswillimply

thatthemultiplicity ofa∈ F as arootof f (p) isatleast thesumof themultiplicities

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sign : R → S will yield Descartes’ rule of signs, and given avaluation v ona ﬁeld K

(which is the samething as ahomomorphism from K to T ) we will recover Newton’s

polygonrule.

Content overview In section 1, we explain the overall idea behind our simultaneous

proof of Descartes’ rule of signs and Newton’s polygon rule. In section 2, we give a

rigorous deﬁnition of hyperﬁelds and a proof of Lemma A and Proposition B. The

above-mentioned interpretation of the multiplicities of roots over the sign hyperﬁeld

isestablishedinsection3,andforthetropicalhyperﬁeldthisisworkedoutinsection4.

InAppendixA,weinvestigatediﬀerentpossiblenotionsof“polynomialalgebra”overa

hyperﬁeldF .Wearguethatwhiletheoldertheoryof“additive-multiplicativehyperrings”

leadstoaratherbadlybehavednotion,thesecondauthor’stheoryoforderedblueprints

furnishes an eﬃcient and satisfying (at least from a categorical perspective) theory of

polynomial algebras over hyperﬁelds. We also discuss how the theory described in the

bodyofthispapergeneralizesneatlytoorderedblueﬁeldswhichsatisfya“reversibility”

axiom.

1. Statementofthe mainresults

Introductiontohyperﬁelds Wealreadymentionedthathyperﬁelds1_are_a_{generalization}

of ﬁelds where addition is allowed to be multi-valued. Somewhat more precisely, in a

hyperﬁeldF additionisreplacedbyahyperoperation ,whichisamap

: F× F −→ P(F )

intothepowersetP(F ) ofF .ThemultiplicationandhyperadditionoperationsonF are

required to satisfy various axioms,the mostnon-obvious of which is thatthere should

be adistinguished neutral element 0 ∈ F such thatfor each x∈ F , there is a unique

−x∈ F suchthat0∈ x(−x).We willgiveamoreprecise deﬁnitioninsection2;for

now wecontentourselveswithsomeexamples.

Thethreemostimportantexamplesofhyperﬁelds,forthepurposesofthispaper,are

thefollowing:

• Every fieldK istautologicallyahyperfieldbydefiningab={a+ b}.

• Thesign hyperﬁeld S consistsofthree elements{0,1,−1} withtheusual

multiplica-tionandhyperadditioncharacterizedbytherules11={1},−1− 1={−1} and

1− 1={0,1,−1}.

• The tropical hyperﬁeld T hasfor itsunderlyingset R∪ {∞}.Multiplication inT is

givenbyadditionof(extended)realnumbers,andhyperadditionisdeﬁnedasfollows:

1 _Marc_Krasner_introduced_hyperrings_and_hyperﬁelds_in_[₁₇_],_and_since_then_many_authors_have

consid-eredmoregeneralversionsofthesenotions.Thehyperﬁeldsconsideredinthispaper,however,willallbe hyperﬁeldsintheoriginalsenseofKrasner.

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if a = b then ab = min(a,b), while aa = {c ∈ R : c a}∪ {∞}. The

hyperinverseofx isequaltox forallx∈ T.

Thefollowinghyperﬁeldswillalsobe usedlateronto givesomeexamplesand

coun-terexamples:

• TheKrasnerhyperﬁeld K consistsoftwoelements{0,1} withtheusualmultiplication

andhyperadditioncharacterized bytherule11={0,1}.

• Theweaksignhyperﬁeld W consistsofthreeelements{0,1,−1} withtheusual

mul-tiplicationandhyperadditioncharacterizedbytherules11=−1 − 1={1,−1}

and1 − 1={0,1,−1}.

• Thephasehyperﬁeld P hasforitsunderlyingsetS1_{∪ {0},}_where_S1₌_{eiθ _{∈ C |}₀

θ < 2π} isthecomplexunitcircle.MultiplicationonP isdeduced fromthe

multipli-cationonC,andhyperadditionis characterizedbythefollowingrules:

– If θ1= θ2+ π,theneiθ1eiθ2 ={0,eiθ1,eiθ2}.

– If θ1< θ2< θ1+ π, theneiθ1eiθ2 ={eiθ|θ1< θ < θ2}.

Remark 1.1.All six of these examples are special cases of a general construction of

hyperﬁeldsas quotientsofﬁelds byamultiplicativesubgroup,whichisdescribedin[5].

LetK beaﬁeldand letG beasubgroup of K×. Thenthe quotientK/G ofK by the

actionof G by (left)multiplication carriesanatural structure ofahyperﬁeld: wehave

(K/G)×= K×/G asanabeliangroupand

[a][b] = [c]c = a+ b for some a∈ [a], b∈ [b]

forequivalenceclasses[a] and[b] in K/G.

For any ﬁeld K we have K = K/{1} and if |K| > 2 then K = K/K×. Similarly, S = R/R>0,P = C/R>0,andW = Fp/(Fp×)2 foranyprimep 7 with p≡ 3 (mod 4).

ThetropicalhyperﬁeldT is alsoaspecialcaseofthequotientconstruction:ifK isany

ﬁeldendowedwithasurjectivevaluation v : K×→ R,thenT = K/v−1(0).

Remark1.2.Thereareexamplesofhyperﬁeldswhichdonotarisefromtheconstruction

giveninRemark1.1;see, forexample,[20].

Rootsandmultiplicities Ifp(T )=n_i=0ciTi isapolynomialwithcoeﬃcientsinaﬁeld

K,anelement a∈ K isarootof p ifandonly ifeitherofthefollowing two equivalent

conditionsissatisﬁed: (1) p(a)= 0,i.e.,ciai= 0.

(2) T − a dividesp(T ), i.e.,there is apolynomialq(T )=n_i=0−1diTi ∈ K[T ] suchthat

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Note that(2) isequivalent totheexistenceofelementsd0,. . . ,dn−1∈ K suchthat

c0=−ad0, ci=−adi+ di−1 for i = 1, . . . , n− 1, and cn = dn−1. (2)

IfF isahyperﬁeld,theninordertodeﬁnewhatitmeanstobearootofapolynomial

over F wewill generalizeconditions(1) and(2) byreplacing sumswithhypersums.

Lemma A.Letc0,. . . ,cn∈ F . Thefollowingare equivalent foranelement a∈ F :

(1) 0∈

ciai.

(2) Thereexistelements d0,. . . ,dn−1∈ F such that

c0=−ad0, ci ∈ (−adi)di−1for i = 1, . . . , n− 1, and cn= dn−1.

Wewrite 0∈ p(a) if(1) issatisﬁed,and p∈ (T − a)q ifq =n−1_i=0 diTi satisﬁes(2).

Wewill giveaproofofLemmaAinsection 2.

Remark1.3.Notethat,unlikethecasewhereF = K isaﬁeld,the“quotient”polynomial

q = n_i=0−1diTi is ingeneral not unique.For example, suppose F = S and let p(T ) =

T3_{− T}2_{− T + 1.}_Then_p_{∈ (T − 1)q for}_{q(T )}_{∈ {T}2_{− 1,}_T2_{+ T}_{− 1,}_T2_{− T − 1}.}

LemmaAmotivatesthefollowingdeﬁnition:

Deﬁnition 1.4.Let c0,. . . ,cn ∈ F . An element a ∈ F is aroot of the polynomialp =

n

i=0ciTi ifitsatisﬁeseitheroftheequivalentconditions(1) or(2).

Wedeﬁne themultiplicitymulta(p) of a asarootofp in termsof asimplerecursion

as follows.

Deﬁnition 1.5.Ifa isnotarootofp,set multa(p)= 0.Ifa isarootofp,deﬁne

multa(p) = 1 + max

multa(q)p∈ (T − a)q

.

NotethatwhenF = K isaﬁeld,multa(p) isjusttheusualmultiplicityofa asaroot

of p.

Remark 1.6. The ideato deﬁne roots of polynomials over hyperﬁelds using (1) is due

to Viro, cf. [24]. However, we believethatLemma A and the deﬁnitionof multa(p) in

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Homomorphisms ofhyperﬁelds

Definition 1.7. Let F1,F2 be hyperfields. A map f : F1 → F2 is called a hyperfield

homomorphism iff (0)= 0,f (1)= 1,f (ab)= f (a)f (b), andf (a+ b)⊂ f(a)f (b) for

alla,b∈ F1.

Example1.8.Here areacoupleofexamplesofhyperﬁeldhomomorphisms.

(1) The functionsign : R → S takingareal numberto its signis ahomomorphism of

hyperﬁelds.

(2) IfK isaﬁeld,amap v : K→ R∪ {∞} iscalleda(Krull)valuation ifv−1(∞)= 0

and for all a,b ∈ K we have v(ab) = v(a)+ v(b) and v(a+ b) min{v(a),v(b)}.

One checks easily that a map v : K → R∪ {∞} is a valuation if and only if the

corresponding mapK→ T is ahomomorphismofhyperﬁelds.

PropositionB.LetK be aﬁeldandf : K → F ahomomorphismtoahyperﬁeld F .Let

p=ciTi beapolynomialoverK andletp =¯ f (ci)Ti thecorresponding polynomial

overF . Then

multb(¯p)

a∈f−1_(b)

multa(p) (1)

for every b ∈ F . Moreover, if _b∈Fmultb(¯p) deg(¯p) and p splits into a product of

linearfactors overK, thenwehave equalityin(1).

Wewillgiveaproofof PropositionBinsection 2.

Remark1.9(Apathologicalexample).IfF isahyperﬁeldandp isapolynomialofdegree

d overF ,itispossibleforthesum_a∈Fmulta(p) toexceedd.Forexample,ifF = W is

theweaksignhyperﬁeld,thenboth1 and−1 aredoublerootsofthequadraticpolynomial

p(T )= T2_{+ T + 1.} _(Indeed,_it_is _immediately_veriﬁed_that₀_{∈ q(1) for} _q_{∈ {p,}_T_{− 1},} 0∈ q(−1) for q∈ {p,T + 1},p∈ (T + 1)(T + 1),andp∈ (T − 1)(T − 1).)

Such“pathological” behavior doesnothappenwhenF is aﬁeldor whenF = K, S,

orT ;inthesecases,_a∈Fmulta(p) d foreverypolynomialp overF byRemarks1.11,

1.13,and1.17below.

Remark1.10(Anevenmorepathologicalexample).Anonzeropolynomialp overa

hyper-ﬁeldF canhaveinﬁnitelymany roots,inwhichcase_a∈Fmulta(p)=∞.Forexample,

takeF = P tobethephasehyperﬁeldandletp(T )= T2_{+ T + 1.}_Then_a_{= e}iθ _is_a_root

ofp forallπ/2< θ < 3π/2.

Remark1.11.Ifp(T )= crTr+ cr+1Tr+1+· · · + cnTn isapolynomial overtheKrasner

hyperﬁeldK,whereweassumethatcr,cn= 0,thenonecheckseasilythatmult0(p)= r

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Remark 1.12.The inequalityprovidedby PropositionBdoes nothold ingeneralifwe

replace K by an arbitrary hyperﬁeld.For example, consider themap f : P → K from

thephasehyperﬁeldtotheKrasnerhyperﬁeldthatsendsallnonzeroelementsofP to1.

ByRemark1.11,1 isaroot ofmultiplicity2 ofthepolynomialT2_{+ T + 1 over}_K._But

when consideredas a polynomialoverP , it hasinﬁnitely manyroots by Remark1.10,

sothesumofthemultiplicitiesofallrootsofT2_{+ T + 1 in}_{P is}_not_bounded_from_above

by2.

It follows from theresults ofthis text, however,thatProposition Bistrue forT or

S in place ofK. In the caseof the tropical hyperﬁeldT , this follows from the unique

factorizationofpolynomialsoverT intolinearterms(Theorem4.1),whichallowsforthe

sameargumentsasinthecaseofaﬁeldtoprovetheinequalityofPropositionB.Inthe

case of the signhyperﬁeld S,the inequalityof Proposition Bcan be establishedalong

the following lines. There is only one proper surjection from S to another hyperﬁeld,

namelythemapf : S→ K thatsends±1 to1.Letp=±Tn_{+ . . .}_{± T}m_be_a_polynomial

over S and p its¯ image polynomial over K. The inequality for b = 0 is clear. Since

mult−1(p)= mult1(p(−T )),wecandeducetheinequalitymult1(p)+ mult−1(p) n− m

from Descartes’sruleofsigns(TheoremC).Sincen− m= mult1(¯p) (cf.Remark1.11),

we obtainthedesiredinequality.

Multiplicities overthe sign hyperﬁeldand Descartes’ rule of signs Let p(T )=ciTi

be apolynomial over the sign hyperﬁeldS, so thatall coeﬃcients are 0, 1 or−1. We

deﬁne thenumberof signchangesin thecoeﬃcientsof p as

σ(p) = #ici =−ci+k= 0 and ci+1=· · · = ci+k−1= 0 for some k 1.

Thefollowing resultwill beprovedinsection 3.

Theorem C.Letp beapolynomial overS. Thenmult1(p)= σ(p).

Remark1.13.Weleaveit asaneasy exercisefor thereaderto verify,using Theorem C

and thefactthat−1 isaroot ofp(T ) if andonly if1 isaroot ofp(−T ),thatifp is a

polynomialoverS then_a∈Smulta(p) deg(p).

AsaconsequenceofTheoremCandPropositionB,weobtainanewproofofDescartes’

rule ofsigns.

Theorem (Descartes’ rule of signs).Let p = ciTi be a polynomial over R and let

p =sign(ci)Ti.Thenthenumberofpositiverealroots ofp (countingmultiplicities)is

at mostσp,with equalityifp splits intoaproductof linearfactors overR. Proof. Since neither _a>0multa(p) nor σ

p changes if we multiply f by a nonzero

real number,we canassume thatf is monic. ByTheorem C, σp= mult1

p. Since sign(a)= 1 ifandonlyifa> 0, PropositionBimplies

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a>0

multa(p) mult1

p = σp,

which establishes the ﬁrstpart of the theorem.The assertionregarding equality when

p splits into a product of linear factors over R follows from Proposition B and

Re-mark1.13.

Remark 1.14.For any polynomial p¯∈ S[T ] there exists a polynomial p ∈ R[T ] with

sign(p)= ¯p such thatthe numberof positive (resp.negative) real roots of p (counting

multiplicities) is equal to mult1(¯p) (resp. mult−1(¯p)), cf. [10]. So the bound given by

our Proposition Bwhen the homomorphism in question is sign : R → S is tight ina

particularlystrongsense.

Itwouldbeinterestingtocharacterizethehyperﬁeldhomomorphismsf : K→ F with

theproperty that forany p over¯ F , there exists apolynomialp∈ K[T ] with f (p)= ¯p

suchthatmultb(¯p) = a∈f−1(b)multa(p) foreveryb∈ F .

Short historical account Descartes stated his rule without proof in the appendix La

géométrie([8])tohisbookDiscoursdelaméthode,whichwaspublishedin1637.Newton

restatedthis formula in 1707, alsowithout aproof. Theﬁrst proof appearsin 1740 in

text Usages de l’analyse de Descartes ([11]) by de Guade Malves. It was reproven by

Gauß ([9]) in 1728, including the addition that the diﬀerence between the number of

positiverealrootsandnumberofsignchangesisalwayseven, whichisoftenmentioned

asapartofDescartes’rule.

TropicalmultiplicitiesandNewton’s polygonrule

Deﬁnition1.15.Given apolynomialp=n_i=0ciTi ofdegreen with ci∈ T, itsNewton

polygon NP(p) isdeﬁnedtobethelower convexhullof{(i,ci) : 0 i n}⊂ R2.(For

simplicity, we assume thatc0 = ∞; this allows us to avoid having to consider vertical

segmentsintheNewton polygon.)

Morevividly,imaginethepoints(i,ci) asnailsstickingoutfromtheplaneandattach

along piece of string with oneend nailed to (x0,y0)= (0,c0) and theother end free.

Rotatethestringcounter-clockwiseuntil itmeetsoneof thenails;thiswill bethenext

vertex(x1,y1) oftheNewtonpolygon.Aswecontinuerotating,thesegmentL1ofstring between(x0,y0) and(x1,y1) willbeﬁxed.Continuingtorotatethestringinthismanner untilthestringcatchesonthepoint(xt,yt)= (n,cn) yieldstheNewtonpolygonofp.

ThusNP(f ) isaﬁnite unionL1,. . . ,Ltof linesegments,eachwith adiﬀerent slope.

We let sj be the negative of the slope of Lj and we denote by λj the length of the

projectionofLj to thex-axis.

Finally,fors∈ R wedeﬁne νs(p) tobe0ifs= sj forallj = 1,. . . ,t,and otherwise

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Example 1.16.Weillustrate these deﬁnitionsinthefollowing example.Let p=ciTi

be themonic polynomialof degree5 withc0= 2, c1= 0, c2= 1,c3=∞, c4 =−1 and

c5= 0.Then theNewton polygoncanbeillustratedas follows:

0 1 2 −1 1 2 3 4 5 (0, c0) (1, c1) (2, c2) (4, c4) (5, c5) L1 L2 L3 i ci k 1 2 3 sk 2 1/3 −1 λk 1 3 1

We displaythe valuesofthe sk andthe λk for theline segmentsL1, L2 and L3 in the

table nextto thegraphic. Thus thefunction νs(p) hasthe valuesν2(p) = 1,ν1/3(p) =

3,ν₋₁(p)= 1,andνs(p)= 0 for alls∈ {2,/ 1/3,−1}.

Thefollowing resultwill beprovedinsection 4.

Theorem D.Letp beapolynomial overT .Forevery s∈ T,wehave mults(p)= νs(p).

Remark1.17.Itfollowsimmediatelyfrom TheoremDthat_a∈Tmulta(p)= deg(p) for

everypolynomialp overT .

Using TheoremDand PropositionB,wededuce:

Theorem (Newton’s polygon rule).Let K be aﬁeld and let v : K → T be a valuation.

Let p= ciTi be a polynomial over T and let p = v(ci)Ti. Let s ∈ T. Then the

number ofrootsa∈ K ofp withv(a)= s (countingmultiplicities)isatmostνs(p),with

equalityif p splits intoaproductoflinear factorsover K.

Proof. Newton’spolygonrulecanbe provenwiththesameargumentasDescartes’rule

of signs, where we rely onTheorem D instead of Theorem C in this case. Namely, by

PropositionBandTheoremD,wehave

a∈v−1_(s)

multa(p) mults(p) = vs(p).

Thus theﬁrstclaim ofthetheorem.Ifp splits intolinearfactors, thenwe deducefrom

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deg(p) =

a∈K

multa(p) mults(p) = deg(p) = deg(p),

andthusequalitythroughout,whichestablishesthesecondclaimofthetheorem.

Remark1.18.If K iscompletewith respectto thevaluationv (i.e., K iscomplete asa

metric space with respect to thedistance function d(a,b) = e−v(a−b)), then v extends

uniquelytoavaluationonanyﬁxedalgebraic closureK of¯ K;cf. [21,ChapterII,Thm.

4.8].Sointhiscase,Newton’spolygonrulecanbeformulatedasfollows: thenumberof

rootsa∈ ¯K ofp withv(a)= s (countingmultiplicities)isequalto νs(p).

Remark 1.19.When K is complete, one often uses Hensel’s Lemma [21, Chapter II,

Lemma 4.6] in conjunction with Newton’s polygon rule to guarantee the existence of

precisely νs(p) roots in K with valuation s. For example, if p has coeﬃcients in the

valuation ring R of K and the reduction of p modulo the maximal ideal of R splits

completelyintodistinctlinearfactors,thenitfollowsfromHensel’sLemmathatp splits

completelyinto linearfactorsoverK.

Remark1.20.It wouldbe interestingto ﬁndother usefulapplications ofPropositionB

besidesDescartes’rule andNewton’s polygonrule.2 Itwouldalsobe interestingto

for-mulate a higher-dimensional version of the theory of multiplicities developed in this

paper.

Relation to the supertropical numbers and the symmetrization of the tropical numbers

Thetropicalhyperﬁeldiscloselyrelatedtothesupertropicalnumbers,whichwere

intro-ducedbyIzhakianin[13].Toexplain,wecanextendtheproductandthehyperaddition

onthetropicalhyperﬁeldT tothewholepowersetP(T) ofT byelementwiseevaluation.

Withtheseoperations,P(T) becomesasemiring.ThesmallestsubsemiringofP(T) that

contains all singletons {a} with a ∈ T consists of all singletons {a} together with all

intervals ofthe form [a,∞], where a variesthrough T . This subsemiringis isomorphic

to Izhakian’s semiring of supertropical numbers. In joint work with Rowen, Izhakian

extends his theory in [14] and [15] to polynomials over the semiring of supertropical

numbers.Thecommon intersections with thecontent ofthis paper are: (a)they

intro-ducetheconceptof aroot, whichagreeswith oursunderthecorrespondence described

above;and(b)theyshow in[14,Lemma5.7] thateverysupertropicalpolynomialhasa

root.

Inthe sameway, the powerset P(S) of thesign hyperﬁeldS is asemiring, and the

smallestsubsemiringwhich containsallsingletons is isomorphicto the symmetrization

of the Boolean semiﬁeld. This semiring, or more precisely its extension to the

sym-metrizationofthetropicalnumbers,areintroducedin[22];alsocf.[1,section3.4].In[1,

2 _Note_added:_the_ﬁrst_author’s_student_Trevor_Gunn_has_recently_found_a_simultaneous_{generalization}_of

Descartes’ruleandNewton’spolygonrulebyapplyingPropositionBtothesignedtropicalhyperﬁeld;see [12] fordetails.

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section 3.6],polynomialsover thissemiring aretreated,including theconceptsofroots

andtheirmultiplicities.Theirnotionofrootscoincideswithours,buttheirdeﬁnitionof

multiplicity forrootsisdiﬀerentfrom ours.

2. Hyperﬁelds

Togivearigorousdefinitionofhyperfields,wefirstdefineabinary hyperoperation on

asetG to beamap

: G× G −→ P(G)

into thepowersetP(G) ofG suchthatab is non-emptyforalla,b∈ G.

The hyperoperation is called commutative if ab = ba for all a,b ∈ G, and

associative if d∈bc ad = d∈ab dc foralla,b,c∈ G.

If isbothcommutativeandassociative,wecandeﬁnethehypersum

n_i=1 aifor

alln 2 anda1,. . . ,an∈ G bytherecursiveformula n

i=1 ai = b∈

n−1i=1 ai ban.

A commutative hypergroup isa set G endowed with a commutativeand associative

binaryhyperoperation andadistinguishedelement0∈ G suchthatforalla,b,c∈ G:

(HG1) 0a= a0={a}. (neutralelement)

(HG2) Thereisauniqueelement −a inG suchthat0∈ a(−a). (inverses)

(HG3) a∈ bc ifandonlyif−b∈ (−a)c. (reversibility)

A hyperﬁeld isa set F togetherwith a binary operation ·, abinary hyperoperation

,and distinguishedelements0 and1 suchthatforalla,b,c∈ F :

(HF1) (F,,0) isacommutativehypergroup.

(HF2) (F\ {0},·,1) is anabeliangroup.

(HF3) a· 0= 0· a= 0.

(HF4) a· (bc)= abac,where a· (bc)={ad|d∈ bc}. (distributivity)

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Proof of LemmaA. The case a= 0 is easy:we have 0∈

ciai = 0· · ·0c0 if

andonlyifc0= 0.Ontheother hand,theconditionsin(2) reduceto

c0= 0, ci∈ 0di−1={di−1} for i = 1, . . . n − 1, and cn= dn−1,

whichcanbe fulﬁlled(uniquely)bydi= ci+1 fori= 0,. . . ,n− 1 ifandonly ifc0 = 0.

Thisestablishesthedesiredequivalencefora= 0.

Ifa= 0,thenbytheverydeﬁnitionofthehypersumofn+ 1 summands,0∈

ciai

ifandonlyifthere isasequence ofelementse1,. . . ,en−1 ∈ F suchthat

e1 ∈ c0c1a, ei ∈ ei−1ciai for i = 2, . . . , n− 1, and 0 ∈ en−1cnan.

Let d0,. . . ,dn−1 ∈ F be the uniqueelements satisfying c0 =−ad0 and ei = −diai+1.

Thentheaboverelationscanberewrittenas

−diai+1 ∈ (−di−1ai)ciai for i = 1, . . . , n− 1, and − dn−1an = −cnan.

(Hereweusethefactthat,by(HG2),0∈ en−1cnan ifandonlyifen−1 =−cnan.)

These relations can be brought into the form in which they appear in (2) by ﬁrst

multiplying each of them by −a−i and then using the reversibility Axiom (HG3) to

exchangethetermsdi and −ci.

Wealsogivethepromised proofofPropositionB:

Proof of PropositionB. Let a1,. . . ,an ∈ K be not necessarily distinct elements such

that(T−ai) dividesp inK[T ].Deﬁneq1= p andfori= 1,. . . ,n,deﬁnethepolynomial

qi+1∈ K[T ] bythepropertythatqi= (T − ai)qi+1 inK[T ].

Toprovethe proposition,assume thatp(a1)= . . . = p(an)= b andthatthere is no

a∈ K such thatf (a)= b andqn+1(a)= 0,i.e., thata1,. . . ,an areallof therootsofp

(countedwithmultiplicities)havingf (ai)= b.

Bythedeﬁnition ofahomomorphism of hyperﬁelds,therelations qi= (T − ai)qi+1

implythatqi∈ (T −b)qi+1overF ,whereqiistheimageofqiunderf .Thusthesequence

oftheqicertiﬁesthatmultb(p) isatleastn.Thisprovestheﬁrstpartoftheproposition.

If p splitsinto linearfactors and _b∈Fmultb(p) deg p, then the ﬁrst assertion of

thepropositionimpliesthat

deg p =

a∈K

multa(p)

b∈F

multb(p) deg p = deg p,

and thus equality holds throughout. Therefore multb(¯p) =

a∈f−1_(b)multa(p) for all

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3. Multiplicitiesoverthesignhyperﬁeld

Ourgoalinthis sectionistoproveTheoremC.

Letp=ciTi beamonicpolynomialoverthesignhyperﬁeldS ofdegreen.Recall

thatthenumberof signchangesin thecoeﬃcientsof p is

σ(p) = #ici =−ci+k= 0 and ci+1=· · · = ci+k−1= 0 for some k 1

.

Theorem3.1. Letp=ciTibeamonicpolynomialofdegreen overS.Thenmult1(p)=

σ(p).

Proof. The maineﬀortoftheproofconsistsinshowingthatifσ(p)> 0 then

σ(p) = 1 + maxσ(q)p∈ (T − 1)q.

Once we have shown this, we can conclude the proof of the theorem by induction on

σ(p). If σ(p)= 0, then0∈ p(1)/ = 1· · ·1 andthus mult1(p)= 0.Ifσ(p)> 0,then 0∈ p(1)= cn· · ·c0 sincethereisasignchange,and

σ(p) = 1 + max{σ(q)p∈ (T − 1)q = 1 + max{mult1(q)p∈ (T − 1)q

= mult1(p),

where we use the inductive hypothesis for the second equality and the deﬁnition of

mult1(p) forthelastequality.

Weproceed with showingthatthemaximumofthevaluesσ(q) with p∈ (T − 1)q is

σ(p)− 1.Let q = diTi be a polynomialover S such thatp∈ (T − 1)q.This means

thatdeg q = deg p− 1 and

d0=−c0, ci∈ −didi−1for i = 1, . . . , n− 1, and dn−1 = cn= 1.

Thestrategyoftheproofistoboundthenumberofsignchangesinq bythenumberof

signchangesinp indecreasingorderof i.

Letσi(p) bethenumberofsignchangesinthesequenceofcoeﬃcientscn,. . . ,ci ofp,

i.e.,

σi(p) = #

k ick =−ck+l+1= 0 and ck+1=· · · = ck+l= 0 for some l 0

.

Letσi(q) bethenumberofsignchangesinthesequenceofcoeﬃcientsdn−1,. . . ,di ofq,

whichisdeﬁnedanalogously toσi(p).

Weclaimthatσi(q) σi(p) foralli= 0,. . . ,n,withσi(q)+ 1 σi(p) ifdi=−ci= 0.

Wewill provethis claimbydescendinginductiononi.Ifi= n,thenσi(q)= σi(p)= 0,

whichproves ourclaiminthiscasesincedn= 0= −cn.

Before explaining the inductive step, we begin with some preliminary observations

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consider.Namely, if0= ci and ci = −ci+1 as well as 0= di anddi = −di+1, then we

haveσi(p)= σi+1(p) andσi(q)= σi+1(q).Thuswedonotchangethevaluesofσi(p) and

σi(q) ifweomitci+1 and di+1 from thesequencescn,. . . ,ci anddn−1,. . . ,di.Therefore

we mayassume without loss of generality thatthis situationdoes not occur.We may

similarlyassumethatc0= 0,sinceotherwised0=−c0= 0 andthusσ0(p)= σ1(p) and

σ0(q)= σ1(q).

Theseassumptionsandtherelationp∈ (T − 1)q havethefollowingconsequencesfor

i= 0,. . . ,n− 1:

(1) Wehaveci+1= 0.Indeed,ifci+1= 0,thenci+1∈ −di+1diimpliesthatdi+1= di.

Butthissituationisexcludedbyourassumptions.

(2) Ifdi+1=−di,thenci+1∈ −di+1di impliesthatci+1 = di=−di+1.

(3) Ifci =−di,thenwehaveci+1= di=−ci.Indeed,ifci+1 = cithenci+1∈ −di+1di

impliesdi+1= di,whichisexcludedbyourassumptions.

Assume that i < n. We prove the inductive step of our claim by considering the

followingfourconstellationsofpossiblevaluesforci,di,anddi+1.(Weindicateusageof

theinductivehypothesisinthefollowingrelationsby“(IH)”.)

Case1: di+1 = −di andci= −di.Inthiscase,weobtain

σi(q) = σi+1(q)

(IH)

σi+1(p) σi(p).

Case2: di+1=−di andci= −di. By(1) and(2),we haveci+1=−di+1= di = ci, and

thus

σi(q) = σi+1(q) + 1

(IH)

σi+1(p) = σi(p).

Case3: di+1 = −di andci=−di.By(3),wehaveci+1 = di=−ci, andthus

σi(q) + 1 = σi+1(q) + 1

(IH)

σi+1(p) + 1 = σi(p).

Case4: di+1 =−di andci=−di.By(3),wehaveci+1 = di=−ci =−di+1,andthus

σi(q) + 1 = σi+1(q) + 2

(IH)

σi+1(p) + 1 = σi(p).

Thisconcludestheproofofourclaim.

Notethatσ(p)= σ0(p) andσ(q)= σ0(q).Sinced0=−c0andq waschosenarbitrarily

withrespectto thepropertyp∈ (T − 1)q,thisshowsthat

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To complete the proof of the theorem,we haveto show thatthere is aq0 with p∈ (T− 1)q0 andσ(q0)+ 1= σ(p).Wedeﬁneq0=diTi asfollows. Letk bethenumber

suchthatc0= . . . = ck =−ck+1,anddeﬁne

di= ci+1 if ci+1= 0 and i > k;

di= di+1 if ci+1= 0 and i > k;

di=−c0 if i k.

Weleavetheeasyveriﬁcationthatp∈ (T − 1)q0andσ(q0)+ 1= σ(p) tothereader.

4. Multiplicitiesoftropicalroots

Our goal inthis section is to prove Theorem D. Our proof is based ona hyperﬁeld

version(Theorem4.1below)oftheso-called“Fundamentaltheoremoftropicalalgebra”

(cf. Lemma4.2).

Letp=ciTi beamonicpolynomialofdegreen overT andleta1,. . . ,an∈ T.We

write p∈(T + ai) if

cn−i∈

e1<···<ei

ae1· · · aei

foralli= 1,. . . ,n.

Theorem 4.1(Fundamentaltheorem forthetropical hyperﬁeld).Letp=n_i=0ciTi be a

monic polynomialof degree n overT .Then:

(1) There is a unique sequence a1,. . . ,an ∈ T, up to permutation of the indices,

suchthatp∈(T + ai).

(2) For everya∈ T, wehaveequalities

multa(p) = #

i∈ {1, . . . , n}a = ai

= vp(a).

The rest of this section is devoted to the proof of Theorem 4.1. The main idea of

theproofistoconsiderpolynomialsoverthetropicalhyperﬁeldT asfunctionsfrom the

tropical semifield R toitself,andto comparethehyperfieldandsemifieldperspectives.

As aset,R = R∪ {∞} isequaltoT ,and theyhavethesamemultiplicationas well:

the product ab∈ R isdeﬁned asthe sumof thecorresponding extendedreal numbers.

ThediﬀerencebetweenR andT appearsintheadditionlaw:thesumoftwoelementsa

andb ofR isdeﬁnedasmin{a,b},whichisanelement ofR,opposedtothesubset ab

of T .

Toavoidconfusionbetweentropicaladditionandusual addition(i.e.,tropical

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T by a,b,c,d andelements of R bya,b,c,d. Given anelement a∈ T, wewrite a if we

consider itas anelementof R.Wekeepthepreviouslyestablishednotations forT , i.e.

the hypersum of a and b is denotedby ab and their product by ab. We denote the

tropical sumof two elements a and b of R by min{a,b} andtheir tropical product by

a + b.Wewritei· a for thei-foldsuma +· · · + a ofa withitself.

Anontrivialpolynomialp=ciTi ofdegreen overT deﬁnes afunction

p : R −→ R b −→ min i=0,...,n ci+ i· b ,

whichwesometimesextendtoafunctionR→ R viap(∞)=∞.Thetrivialpolynomial

yieldsthetrivialfunction b → ∞.

We say that two polynomials p =ciTi and q = diTi over T are functionally

equivalent,denotedp = q, iftheydeﬁne the samefunctionR → R.We call afunction

p : R→ R asaboveatropical(polynomial)function anddenoteitbyp = min{ci+ i·T }.

Thedegreeof p isthedegreeofp andp ismonic ifp ismonic.Notethatbothnotionsare

independentofthechoiceoftherepresentingpolynomialp.Notefurtherthattheset of

tropicalfunctionsinheritsthestructureofasemiringfromR byaddingandmultiplying

functionsvaluewise.

Itiswell-knownthateverytropicalfunctionfactorsuniquelyinto aproductoflinear

functions.This resultissometimesreferredto asthe “fundamentaltheoremof tropical

algebra”,anditwasﬁrstprovenin[6,Thm.11];seealso[1,Thm.3.43].

Lemma4.2(Fundamentaltheoremoftropicalalgebra).Foreverymonictropicalfunction

p = min{ci + i· T } of degree n, there is a unique sequence a1,. . . ,an ∈ R, up to a

permutationof indices,suchthat p =_i=1n min{T,ai} astropical functions.

Thesecondequalityinpart2ofTheorem4.1followsfromtheusualargumentsinthe

theoryofNewtonpolygons;inparticular,wehavethefollowingwell-knownfact(see[4]

or[6,section9],forexample,forproofs):

Lemma4.3.Letp=ciTibeamonicpolynomialofdegreen overT andleta1,. . . ,an ∈

T besuchthat p =n_i=1min{T,ai}.Leta∈ R. Then#{i|a= ai}= vp(a).

Therestoftheproof ofTheorem4.1 isnovel.Part(1) follows immediatelyfrom the

followingproposition,coupledwithLemma4.2.

Proposition4.4. Letp beamonicpolynomialofdegree n overT andleta1,. . . ,an ∈ T.

Thenp∈(T + ai) if andonlyif p =min{T,ai} astropical functions.

Proof. Letp=ciTi andassumethata1 · · · an.Wedeﬁne

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which can be thought of as the i-th elementary symmetric polynomial evaluated at (a1,. . . ,an) (withrespectto thetropicaladditionfromR,notthehyperadditionofT ).

Thus n i=1 min{b, ai} = min i=0,...,n{si+ ib}.

Therelationp∈(T + ai) meansthatcn−i si foralli= 1,. . . ,n,withequalityifthe

minimumoccurs onlyonceamong thetermsae1· · · aei with1 e1<· · · < ei n.This

is thecaseifandonlyifai< ai+1.

Webeginwiththeproofthatp∈(T + ai) impliesp =

min{T,ai}.Forb∈ T,we

have

p(b) = min

i=0,...,n{ci+ ib} mini=0,...,n{si+ ib} = n

i=1

min{b, ai}.

In order to verify the reverse inequality, we choose some a0 min{b,a1} and deﬁne

an+1 = ∞. Then ak b < ak+1 for some k ∈ {0,. . . ,n}. Since ak < ak+1, we have

cn−k= sk,asnotedbefore.Therefore

p(b) = min i=0,...,n{ci+ ib} cn−k+ (n− k)b = sk+ (n− k)b = n i=1 min{b, ai}.

This concludestheproofthatp =min{T,ai}.

Wecontinuewiththereverseimplicationandassumethatp =min{T,ai}.Weneed

to showfork = 1,. . . ,n thatcn−k sk,withequality ifak < ak+1.Chooseb∈ T such

thatak b ak+1,where wesetan+1=∞ asbefore.Then

min i=0,...,n{ci+ ib} = p(b) = n i=1 min{b, ai} = a1+· · ·+ak+ b + · · · + b n−k times = sk+ (n−k)b.

It follows, inparticular, that c_n−k sk. If ak < ak+1, then p(b) = sk+ (n− k)b for

inﬁnitelymanyb.Thisisonlypossibleifc_n−k= sk.

Weareleftwithprovingtheﬁrstequalityinpart(2) ofTheorem4.1.Asaﬁrststep,

we willprovethefollowingfact.(Tomakesense ofthecasen= 1,wedeﬁne theempty

product ofpolynomialsoverT as{0}.)

Lemma 4.5. Let p be a polynomial over T and let a1,. . . ,an ∈ T be such that p ∈

n

i=1(T + ai). Ifp∈ (T + an)q for apolynomialq over T ,then q∈

n−1

i=1(T + ai).

Proof. Notethatthehypothesesofthepropositionimplythatp ismonicofdegreen 1

and that q is monic ofdegree n− 1. Weprove theresult byinduction on n.If n = 1,

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Let n > 1. By part (1) of Theorem 4.1, q ∈ n_i=1−1(T + a_i) for some sequence

a₁,. . . ,a_n−1∈ T.Thismeansthat

di ∈

1e1<···<en−i−1<n

(ae1+· · · + aen−i−1)

foralli= 0,. . . ,n− 2.Thusp∈ (T + an)q impliesthat

ci ∈ di−1dian

⊂

1e1<···<en−i<n

(a_e1· · · a_e_n−i)

1e1<···<en−i−1<n

(a_e1· · · a_e_n−i−1an)

=

1e1<···<en−in

(a_e1· · · a_e_n−i)

fori= 1,. . . ,n−1,whereweseta_n= an.Alsoc0= d0an=ai,andthusp∈

(T +a_i). Bytheuniquenessofa1,. . . ,ansuchthatp∈(T +ai) (bypart(1) ofTheorem4.1),we

concludethatthere isapermutationσ∈ S_n−1 suchthata_i= aσ(i)for i= 1,. . . ,n− 1.

Thusq∈n−1_i=1(T + ai),asclaimed.

In order to complete the proof of Theorem 4.1, consider a monic polynomial p =

ciTi ofdegreen overT withp∈(T + ai) andleta∈ T.Then∞∈ p(a) ifandonly

ifthe minimum appears twiceamong theterms ci+ i· a for i= 0,. . . ,n. This means

thatthefunctionp : R→ R hasachangeof slopeat a,whichisthecaseifand onlyif

a∈ {a1,. . . ,an}.

We prove that multa(p) = #{i|a = ai} by induction on the latter quantity. If

#{i|a = ai} = 0, then a ∈ {a/ 1,. . . ,an} and ∞ ∈ p(a)./ Thus multa(p) = 0, as

de-sired.

If #{i|a = ai} > 0, then a ∈ {a1,. . . ,an} and ∞ ∈ p(a). After relabeling the

indices,wecanassumethata= an.Foreverypolynomialq overT with p∈ (T + an)q,

Proposition4.5showsthatq∈n−1_i=1(T + ai).Thustheinductivehypothesisappliesto

q andyields multa(p) multa(q) + 1 = # i∈ {1, . . . , n − 1}a = ai + 1 = #i∈ {1, . . . , n}a = ai .

Bydeﬁnition, multa(p) = 1+ max{multa(q)|p ∈ (T + a)q}. By LemmaA, there is a

polynomialq0 suchthatp∈ (T + a)q0. Sinceq wasarbitrary,theﬁrst inequalityinthe

displayedequation isanequality.Thisconcludestheproofof Theorem4.1.

Appendix A. Polynomialalgebrasoverhyperﬁelds

Upto this point, we have consideredpolynomials over ahyperﬁeld F asformal

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tomakesenseofsuchexpressionsaselementsofa“polynomialalgebra”overF ,andhow

thedeﬁnitionsof rootsandtheirmultiplicitiestakeamoreconventionalforminsucha

formulation.

Infact,wewillconsider twocandidatesforthepolynomialalgebraoverahyperﬁeld:

as a“additive-multiplicative hyperring” withmulti-valued multiplicationand addition,

or as anordered blueprint. Weargue thatthesecond ofthese alternatives is themore

naturalandlesspathologicalone.

A.1. Polynomial hyperrings

LetF beahyperﬁeld.ThesetPoly(F )={ciTi|ci∈ F } ofallpolynomialsoverF

can be naturallyendowed with two hyperoperations and

, whichare deﬁned for

polynomials p=ciTi andq = diTi as pq = eiTiei∈ cidi , p

q = eiTiei∈

k+l=i ckdl .

TheseoperationsturnPoly(F ) intoanadditive-multiplicativehyperringwhichhasbeen

consideredin[7],[16],andotherpublications.

Let a ∈ F , and let p= ciTi and q = diTi be polynomials over F . Then p∈

(T − a)

q ifand onlyifn= deg p= deg q + 1 and

c0=−ad0, ci∈ (−adi)di−1for i = 1, . . . , n− 1, and cn = dn−1.

This means thatthe relationp∈ (T − a)q, as introduced insection 1,is equivalent to

therelationp∈ (T − a)

q stemmingfromthehypermultiplication ofpolynomialsover

F .

Similar to the case of the hypersum of a hyperﬁeld, we deﬁne n-fold products of

polynomials overF bytherecursiveformula

n

i=1 pi = q∈

n−1 i=1pi q

pn.

In the case of the tropical hyperﬁeld T , the relation p∈ (T + ai) from section 4

is equivalenttop∈

n_i=1(T + ai).Indeed, bymultiplyingoutalllinearterms,weﬁnd

thatp∈

n_i=1(T + ai) isequivalent top=

ciTi being monicofdegreen suchthat

c_n−i ∈

1e1<···<ein

ae1· · · aei

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In spite of these appealing interpretations of the relations p ∈ (T + a)q and p ∈

(T + ai), we view the (additive-multiplicative) polynomial hyperring Poly(F ) as an

objectoflimitedutilityduetothefollowing twodeﬁciencies.

A.2. Deﬁciency#1:polynomial hyperrings arenot associative

Thehypermultiplication ofapolynomialhyperringfails tobe associativeingeneral.

Thisis,forinstance,thecaseforthepolynomialalgebraPoly(S) overthesignhyperﬁeld,

asthefollowingexample(dueto ZiqiLiu,cf. [18])shows:

While (T− 1)

(T− 1)

(T + 1) = T2− T + 1

(T + 1) = T3+ aT2+ bT + 1a, b∈ S, wehave (T− 1)

(T− 1)

(T + 1) = (T − 1)

T2+ aT− 1a∈ S = T3+ aT2+ bT + 1a =−1 or b = −1.

This means, in particular, that n-fold products

_i=1n pi of linear polynomials pi ∈

Poly(S) dependontheorder ofthepi.

RemarkA.1.Notethatthisexamplealsoshowsthatwecannotdeﬁnethemultiplicities

ofrootsinanaïvewayintermsoffactorizationsintolinearfactors:p= T3+ T2+ T + 1

isanelementof(T− 1)

(T− 1)

(T + 1),butp(1) doesnotcontain0.

Remark A.2.Liu also shows in [18] that hypermultiplication in Poly(T ) is

non-associative. Note, however, that Theorem 4.1 implies that the hyperproduct of linear

polynomials overT is associative, andthus

n_i=1(T + ai) isindependent oftheorder

ofthefactors.

Remark A.3.We could overcome Deﬁciency #1 by extending the hyperproduct of

polynomials to certain sets of polynomials inthe following way. For a ﬁnite sequence

C0,. . . ,Cn ⊂ F ofsubsetsofF ,wedenotebyCiTithesetofpolynomialsp=ciTi

withcoeﬃcientsci ∈ Ci. Wedeﬁne

(CiTi)(DiTi) =

k+l=iCkDl

Ti_,

which recovers the hyperproduct (ciTi)

(

diTi) = ( CiTi)( DiTi) in the

caseof singletons Ci ={ci} and Di ={di}. Withthese conventions, is associative,

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n

i=1 pi = diTi di∈

j1+···+jn=i n k=1 ck,jk , forpi=ci,jTj.

We will notpursuethis line ofthought any further;note,however, that appears

implicitlyinourproposedsolutionusingorderedblueprints.Namely,q∈

n_i=1piifand

onlyifqn_i=1pi intheassociated orderedblueprint;cf.sectionA.8.

A.3. Deﬁciency#2:polynomial hyperrings arenotfree

Polynomialhyperringsfailtosatisfytheuniversalpropertyofafreealgebra.Infact,it

appearstobethecasethatneitherthecategoryofhyperringsnorasuitablecategoryof

(non-associative)additive-multiplicativehyperringspossessfreealgebrasingeneral.Here

we assumethatamorphism ofadditive-multiplicativehyperringsisamap f : R1→ R2

thatpreserves0 and1 andsatisﬁesf (ab)⊂ f(a)f (b) andf (a

b)⊂ f(a)

f (b).

For instance, we can extend the identity map K → K of the Krasner hyperﬁeld

to diﬀerent morphisms K[T ] → K that map T to 1. One example is the morphism

f1: K[T ]→ K thatmaps everynonzeropolynomialp tof (p)= 1. Anotherexampleis

the morphism f0 : K[T ] → K for which f0(p) = 1 if and only ifp is a monomial, i.e.

p= Tn _for_some_n_0.

A.4. Towards freealgebras

One way to incorporatefree (and associative)algebras over hyperﬁeldsmight be to

develop a theory of “partial hyperrings”, as considered in [2], which allows for such

objects. Inthisappendix,however,wewill usethemoregeneraland alreadydeveloped

theoryoforderedblueprintstoproducefreealgebraswhichsatisfythedesireduniversal

property. Weremark thatonecouldmostlikelyalso developasimilar theorybased on

Rowen’snotionofsystems (cf.[23]),whichissimilartothatofanorderedblueprint.

Inlayman’sterms,thepassagefrom hyperﬁeldsto orderedblueprintsconsists

essen-tiallyinanexchangeofsymbols:therelationsc∈ ab inahyperﬁeldF getreplacedby

therelationsc a+ b intheassociatedordered blueprint.Underthehood,thesymbol

refers toapartialorderthatisdeﬁnedonthegroupsemiringB+_{= N[F}×_].

We now outlinethe deﬁnitionofordered blueprintsand indicatehow theyallow for

freealgebras overhyperﬁelds;formoredetails,wereferthereaderto[3] and [19].

A.5. Ordered blueprints

An ordered semiring is a commutative (and associative) semiring R with 0 and 1

together with a partial order that is additive and multiplicative, i.e. a b implies

a+ c b+ c and ac bc for all a,b,c ∈ R. Given a set S = {ai bi} of relations

on R,we saythat S generates thepartial order on R if is the smallestadditiveand

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Anorderedblueprint isanorderedsemiringB+togetherwithamultiplicativesubset

B• of B+ thatcontains0 and 1 and thatgenerates B+ as asemiring. Wewrite B for

an ordered blueprint and refer to its ambient semiring by B+ _and _to _its _underlying

monoid by B•. A morphism f : B1 → B2 of ordered blueprints is an order-preserving homomorphismf : B+₁ → B₂+ ofsemiringssuchthatf (B₁•)⊂ B₂•.

Example A.4. The tropical semiﬁeld R can be considered as the ordered blueprint B

withB•= B+_{= R whose}_partial_order_satisﬁes _a_{b if}_and_only_if_a_{+ b}_{= b.}

For the purpose of this appendix, we invite the reader to think of R as the

max-times-algebra R₀, incontrast to themin-plus-algebra R∪ {∞} usedinthemain part

of thispaper. The negativelogarithm −log : R₀ → R∪ {∞} deﬁnes anisomorphism

ofsemiringsbetweenthesetwomodelsforR.Notethat agreeswiththenaturalorder

onR0 andwith thereversed natural orderonR∪ {∞}.

A.6. Hyperﬁeldsas orderedblueprints

TheincarnationofahyperﬁeldF asanorderedblueprintB isasfollows.Itsambient

semiring is thegroup semiringB+ = N[F×], its underlying(multiplicative) monoid is

B•= F ,anditspartialorderisgeneratedbyallrelationsoftheformc a+ b whenever

c ∈ ab in F . We illustrate this in more detailfor the main examples of hyperﬁelds

whichappearinthispaper.

A.6.1. Fields

Given a ﬁeld K, the associated hyperaddition is deﬁned as ab = {a+ b}. This

yieldstheordered blueprintB withambientsemiringB+_{= N[K}×_],_underlying_monoid

B•= K,andpartialorder thatisgeneratedby

c a + b whenever c = a + b in K.

A.6.2. Thetropical hyperﬁeld

As with the tropical semiﬁeld R, we adoptthe multiplicative notation from

Exam-pleA.4,i.e.,weidentifytheelementsofthetropicalhyperﬁeldwithR0and,byabuseof

notation,use theletterT forthe associatedordered blueprint,which canbedescribed

explicitly as follows. The ambientsemiring of T is the group semiring T+ _{= N[R}

>0]

generated bythe multiplicativegroupof positive real numbers,theunderlying monoid

isT•= R₀,andthepartialorderis generatedbytherelations

c a + b whenever c = max{a, b} or c a = b in R₀.

Note that the semiring T+ _is _{not idempotent,}_in _contrast _to _the _tropical _semiﬁeld _R.

Rather, it isa subsemiringof thegroup ringZ[R>0]. Theconnection to R is given by

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linearly toan order-preservingsurjectionf : T+ = N[R>0]→ R0 = R

+

ofsemirings,

i.e., f isamorphism oforderedblueprints.

A.6.3. The sign hyperﬁeld

As anordered blueprint,thesignhyperﬁeld S consistsoftheambientsemiringS+₌

N[{1,−1}],theunderlyingmonoid S• ={0,1,−1},and thepartial order generated by therelations

1 1 + 1, 1 1 − 1, and 0 1 − 1.

Note that0 and1− 1 aredistinct elementsinS+_.

A.7. Free algebras

LetB beanorderedblueprintwithambientsemiringB+,underlyingmonoidB•,and

partial order.Thefreealgebra B[T ] over B consistsof theambientsemiring

B[T ]+ = n i=0 riTiri∈ B+ ,

with respect to theusual additionand multiplication rulesfor polynomials,the

under-lying monoid

B[T ]• = { aTi| a ∈ B•}

of monomials inB+_{[T ] with} _coeﬃcients _in_B•_, _and_the _partial _order _generated _by _the

relations

rTn _sTn _whenever _r _s

forr,s∈ B+_._The_universal_property _for_{B[T ] is}_as _follows,_cf._[₁₉_,_Lemma_5.5.2].

Lemma A.5.Foreverymorphismof orderedblueprintsf : B+ → C+ andeveryelement

a ∈ C, there is a unique morphism of ordered blueprints g : B[T ] → C such that

g(r)= f (r) forr∈ B+ _and_{g(T )}_{= a.}

Example A.6.A typicalelement of thefreealgebraT [T ] over thetropicalhyperﬁeldis

of the form riTi where ri ∈ N[T×] is aformal sum ri =

ak of tropical numbers

ak ∈ T×.Forexample,wehave

T2+ T + 1 T2+ T + T + 1 = (T + 1)2

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AtypicalelementofS[T ] isoftheformriTiwhereri∈ N[{1,−1}] isaformalsum

oftheformri= 1+· · · + 1− 1− · · · − 1. Forexample,wehave

T2− 1 T2+ T− T − 1 = (T + 1)(T − 1) since0 T − T ,butequalitydoesnothold inS[T ]+.

A.8. Polynomialhyperrings, revisited

LetF beahyperﬁeldandB theassociatedorderedblueprint.Theneverypolynomial

ciTioverF istautologicallyanelementofthesemiringB[T ]+= N[F×].Thisidentiﬁes

Poly(F ) withasubsetofB[T ]+,whichcanbe recoveredfrom B[T ] asfollows.

LetB be anordered blueprint. A polynomial over B is an element of B[T ]+ _of _the

form p = ciTi with ci ∈ B•. We denote by Poly(B) the subset of polynomials in

B[T ]+.

IfB istheorderedblueprintassociatedwithahyperﬁeldF ,thenPoly(F )= Poly(B)

assubsetsofB[T ]+_._Moreover,_we_obtain_the_following_{reinterpretation}_of_the

hyperad-ditionandhypermultiplication ofpolynomialsoverF :

p1p2 = q∈ Poly(B)q p1+ p2 , p1

p2 = q∈ Poly(B)q p1· p2 ,

wherep1+ p2andp1· p2 are,respectively,thesumandproductofp1andp2aselements ofB[T ]+_._In_other _words, _for_p

1,p2,q∈ Poly(F )= Poly(B) we haveq∈ p1p2 ifand onlyifq p1+ p2 andq∈ p1

p2 ifandonlyifq p1· p2.

A.9. Rootsofpolynomials overordered blueprints

Toclosethecircleofideas,wereformulatethenotionsofrootsandtheirmultiplicities

in our newly developed formalism and then extend these notions to a more general

class of ordered blueprintsthan hyperﬁelds.For this purpose,we introduce thenotion

of a pasture, which is an algebraic structure closely connected to the ‘foundation’ of

amatroid, cf. [3]. There are several equivalent deﬁnitionsof pastures. Inthis text, we

realizethemasaparticulartypeoforderedblueprints.

Webeginwith somepreliminarynotions.Wedenote byB× thegroupof

multiplica-tively invertible elements of B. An ordered blue ﬁeld isanonzero ordered blueprint B

suchthatB = B×∪ {0}. Anordered blueprintB isreversible if itcontainsanelement

 with 2 _{= 1 such} _that_every_relation _a _b_{+ r where} _a,_b _{∈ B}• _and _r _{∈ B}+ _implies

b a+ r. Asshownin[19, Lemma5.6.34], is uniquelydeterminedbythisproperty

and for everyelement a∈ B• there is aunique element b ∈ B• (namely b = a) such

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Deﬁnition A.7.A pasture is a reversible ordered blue ﬁeld B whose partial order is

generated by relationsofthe form c a+ b witha,b,c ∈ B andsuch thatthenatural

map N[B×]→ B+ _is_bijective.

NotethattheorderedblueprintB associatedtoahyperﬁeldF isapasture.Clearly,B

isanorderedblueﬁeld.ThereversibilityAxiom(HG3)forF impliesthatB isreversible.

The last property follows from the fact that the partial order is generated by the

relationsc a+ b for whichc∈ a+ b inF .

Weextendthenotionsofrootsandtheirmultiplicitiestopolynomialsfromhyperﬁelds

to pastures.

DeﬁnitionA.8.LetB beapasture,leta∈ B•,andletp=ciTi beapolynomialover

B. Letp(a) denotetheelementciai ofB+.Then a isaroot ofp if 0 p(a).

Ifa isnotarootofp,wesaythatthemultiplicitymulta(p) ofa is0.Ifa isarootof

p,wedeﬁne multa(p) = 1 + max multa(q)p (T + a)q .

LemmaAgeneralizestopasturesB,withthesameproof.Namely,a∈ B•istheroot

ofapolynomialp∈ Poly(B) ifandonlyifthereisaq∈ Poly(B) suchthatp (T +a)q.

PropositionBalsogeneralizestopastures,withthesameproof.LetB betheordered

blueprintassociated with aﬁeldK (cf. sectionA.6.1)and f : B→ C a morphismto a

pasture C.Let p=ciTi ∈ Poly(B) and denote by p = f (ci)Ti the image of p in

Poly(C). Thenforallb∈ C• wehave

multb(p) a∈B•_with f (a)=b multa(p). References

[1]FrançoisLouisBaccelli,GuyCohen,GeertJanOlsder,Jean-PierreQuadrat,Synchronizationand Linearity,WileySeriesinProbabilityandMathematicalStatistics:ProbabilityandMathematical Statistics,JohnWiley&Sons,Ltd.,Chichester,1992.

[2]Matthew Baker, Nathan Bowler, Matroids over partialhyperstructures, Adv.Math. 343 (2019) 821–863.

[3]MatthewBaker,OliverLorscheid,Themodulispaceofmatroids,Preprint,arXiv:1809.03542,2018. [4] Bill Casselmann, Newton polygons. Notes, www.math.ubc.ca/~cass/research/pdf/Newton.pdf,

2018,versionfromSeptember1,2018.

[5]AlainConnes,CaterinaConsani,Thehyperringofadèleclasses,J.NumberTheory131 (2)(2011) 159–194.

[6]R.A.Cuninghame-Green,P.F.J.Meijer,Analgebraforpiecewise-linearminimaxproblems,Discrete Appl.Math.2 (4)(1980)267–294.

[7]B.Davvaz,A.Koushky,Onhyperringofpolynomials,Ital.J.PureAppl.Math.15(2004)205–214. [8]RenéDescartes,Lagéométrie.AppendixtoDiscoursdelaméthode,1637.

[9]FriedrichCarlGauß,BeweiseinesalgebraischenLehrsatzes,J.ReineAngew.Math.3(1828)1–4. [10]DavidJ.Grabiner,Descartes’ruleofsigns:anotherconstruction,Am.Math.Mon.106 (9)(1999)

(27)

[11]JeanPauldeGuadeMalves,Usagesdel’analysedeDescartes,1740.

[12]Trevor Gunn,A Newton polygonrule for formally-real valuedﬁelds andmultiplicities over the signedtropicalhyperﬁeld,Preprint,arXiv:1911.12274,2019.

[13]ZurIzhakian,Tropicalarithmeticandmatrixalgebra,Commun.Algebra37 (4)(2009)1445–1468. [14]ZurIzhakian,LouisRowen,Supertropicalalgebra,Adv.Math.225 (4)(2010)2222–2286.

[15]ZurIzhakian, LouisRowen,Supertropicalpolynomialsandresultants,J.Algebra324 (8)(2010) 1860–1886.

[16]Sanja Jančic-Rašović, About thehyperring of polynomials, Ital. J. Pure Appl.Math.21 (2007) 223–234.

[17]Marc Krasner, Approximation des corps valués complets de caractéristique p = 0 par ceux de caractéristique0,in:Colloqued’algèbresupérieure,tenuàBruxellesdu19au22décembre1956,in: CentreBelge deRecherchesMathématiques, ÉtablissementsCeuterick/LibrairieGauthier-Villars, Louvain/Paris,1957,pp. 129–206.

[18]ZiqiLiu,Afewresultsonassociativityofhypermultiplicationsinpolynomialhyperstructuresover hyperﬁelds,Preprint,arXiv:1911.09263,2019.

[19] OliverLorscheid,Blueprintsandtropicalschemetheory,Lecturenotes,http://oliver.impa.br/2018 -Blueprints/versions/lecturenotes180521.pdf,2018,versionfromMay21,2018.

[20]Ch.G.Massouros,Methodsofconstructinghyperﬁelds,Int.J.Math.Math.Sci.8 (4)(1985)725–728. [21]Jürgen Neukirch, Algebraic Number Theory, Grundlehren der Mathematischen Wissenschaften,

vol. 322,Springer-Verlag,Berlin,1999.

[22]MaxPlus,Linearsystemsin(max,+)-algebra,in:Proceedingsofthe29thConferenceonDecision andControl(Honolulu),1990.

[23]Louis Rowen, An informal overview of triples and systems,in: Rings, Modules and Codes, in: Contemp.Math.,vol. 727,Amer.Math.Soc.,Providence,RI,2019,pp. 317–335.

[24]Oleg Ya. Viro,Onbasicconceptsoftropical geometry,Proc.SteklovInst.Math.273 (1)(2011) 252–282.

Descartes' rule of signs, Newton polygons, and polynomials over hyperfields

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