Strong bi-homogeneous Bézout theorem and its use in effective real algebraic geometry

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effective real algebraic geometry

Mohab Safey El Din, Philippe Trebuchet

To cite this version:

Mohab Safey El Din, Philippe Trebuchet. Strong bi-homogeneous Bézout theorem and its use in

effective real algebraic geometry. [Research Report] RR-6001, INRIA. 2006, pp.46. �inria-00105204v4�

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inria-00105204, version 4 - 23 Oct 2006

a p p o r t

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Thème SYM

Strong bi-homogeneous Bézout theorem and its use

in effective real algebraic geometry

Mohab Safey El Din — Philippe Trébuchet

N° 6001

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Mohab SafeyEl Din

∗

, PhilippeTrébu het

†

ThèmeSYMSystèmessymboliques ProjetsSALSA

Rapportdere her he n°6001O tobre200643pages

Abstra t: Let

(f

1 , . . . , f

s

)

be a polynomial family in

Q[X

1 , . . . , X

n

]

(with

s 6 n − 1

) of degree bounded by

D

. Suppose that

hf

1 , . . . , f

s

i

generates aradi al ideal, and denes a smoothalgebrai variety

V ⊂ C

n

. Consideraproje tion

π : C

n

_{→ C}

. Weprovethat the degree of the riti al lo us of

π

restri tedto

V

is bounded by

D

s

_{(D − 1)}

n

−s

n

−s

. This resultisobtainedintwosteps. Firstthe riti alpointsof

π

restri tedto

V

are hara terized asproje tionsof thesolutionsof Lagrange'ssystemforwhi h abi-homogeneousstru ture isexhibited. Se ondlyweproveabi-homogeneousBézoutTheorem,whi hboundsthesum ofthedegreesoftheequidimensional omponentsof theradi alofanidealgenerated by a bi-homogeneous polynomial family. This result is improvedwhen

(f

1 , . . . , f

s

)

is aregular sequen e. Moreover,weuse Lagrange'ssystemto designan algorithm omputing at least one point in ea h onne ted omponent of a smooth real algebrai set. This algorithm generalizes, to the non equidimensional ase, the one of Safey El Din and S host. The evaluation of the output size of this algorithm gives newupperbounds on the rst Betti numberofasmoothrealalgebrai set. Finally, weestimateits arithmeti omplexityand provethatintheworst asesitispolynomialin

n

,

s

,

D

s

_{(D −1)}

n−s

n

−s

andthe omplexity ofevaluation of

f

1 , . . . , f

s

.

Key-words: omputeralgebra,polynomialsystemsolving,ee tivereal algebrai geom-etry

∗

Mohab.Safeylip6.fr

†

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Résumé : Soit

(f

1 , . . . , f

s

)

une famille depolynmes dans

Q[X

1 , . . . , X

n

]

(où

s 6 n − 1

) dedegrébornépar

D

. Onsupposeque

hf

1 , . . . , f

s

i

engendreunidéalradi al,etdénitune variétéalgébrique lisse

V ⊂ C

n

. Considérons une proje tion

π : C

n

_{→ C}

. On prouveque ledegrédulieu ritiquede

π

restreinte à

V

estborné par

D

s

_{(D − 1)}

n

−s

n

−s

. Cerésultat estobtenuendeux temps. Toutd'abord,on ara térise lespoints ritiquesde

π

restreinte à

V

omme proje tions des solutions du système de Lagrange pour lequel on exhibe une stru turebi-homogène.Puis,onprouveunethéorèmedeBézoutbi-homogène,quibornela sommedesdegrésdes omposanteséqui-dimensionnellesduradi ald'unidéalengendrépar unefamilledepolynmesbi-homogènes. Cerésultatestaméliorédansle asoù

(f

1 , . . . , f

s

)

est une suite régulière. De plus, on utilise la formulation Lagrangienne pour dé rire un algorithme al ulantau moins un point par omposante onnexe d'une variétéalgébrique réelle lisse. Cet algorithme généraliseau as non équi-dimensionnel eluide Safey El Din et S host. L'estimation de la taille de la sortie de notre algorithme donne de nouvelles (et meilleures)bornes sur lepremier nombrede Betti d'unevariété algébriqueréelle lisse. Finalement, onmontre qu'uneinstan eprobabilistede notrealgorithme est de omplexité polynomiale en

n

,

s

,

D

s

_{(D − 1)}

n

−s

n

n−s

et la omplexitéd'évaluation de

f

1 , . . . , f

s

. Mots- lés: al ulformel,résolutiondesystèmespolynomiaux,géométriealgébriqueréelle ee tive

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1 Introdu tion

Considerpolynomials

(f

1 , . . . , f

s

)

in

Q[X

1 , . . . , X

n

]

(with

s 6 n − 1

)ofdegreeboundedby

D

. Suppose that this polynomial family generates a radi al ideal and denes a smooth algebrai variety

V ⊂ C

n

. The oreofthispaperistogiveanoptimalboundonthedegree ofthe riti allo usofaproje tion

π : C

n

_{→ C}

restri tedto

V

andtoprovideanalgorithm omputingatleastonepointin ea h onne ted omponentofthereal algebrai set

V ∩ R

n

whoseworst ase omplexityispolynomialin thisbound.

Motivation and des ription of the problem. Sin e omputing riti al pointssolves the problem of algebrai optimization, it hasmany appli ations in hemistry, ele troni s, nan ialmathemati s(see[15℄foranon-exhaustivelistofproblemsandappli ations). More traditionally, omputationsof riti alpointsareusedinee tiverealalgebrai geometryto omputeatleastonepointinea h onne ted omponentofarealalgebrai set. Indeed,every polynomialmappingrestri tedtoasmooth ompa trealalgebrai setrea hesitsextremaon ea h onne ted omponent. Thus, omputingthe riti allo usofsu hamappingprovides atleastonepointin ea h onne ted omponentof asmooth ompa trealalgebrai set.

In[22,25,26,9,10℄,theauthors onsiderthehypersurfa edenedby

f

2

1 + · · · + f

s

2 = 0

tostudytherealalgebrai set

V ∩ R

n

. Theproblemisredu ed,viaseveralinnitesimal de-formations,toasmoothand ompa tsituation.Su hte hniquesyieldalgorithmsreturning zero-dimensionalalgebrai setsen odedbyrationalparameterizationsofdegree

O(D)

n

(the best bound is obtained in [9, 10℄ and is

(4D)

n

). Similarte hniques basedon theuse of a distan e fun tion to ageneri pointand a singleinnitesimal deformation(see [34℄) yield thebound

(2D)

n

ontheoutput.

Morere ently,otheralgorithms,avoidingthesumofsquares(andtheasso iatedgrowth ofdegree)havebeenproposed(see[2,35,7,4,5,38,37℄). The ompa tnessassumptionis droppedeitherby onsideringdistan efun tionsandtheir riti allo us(see[2,35,7,6℄)or byensuringpropernesspropertiesofsomeproje tionfun tions(see[37℄). Thesealgorithms omputethe riti allo iofpolynomialmappingsrestri tedtoequidimensionalandsmooth algebrai varietiesofdimension

d

, denedbypolynomialsystemsgeneratingradi alideals. Indeed,undertheseassumptions, riti alpoints anbealgebrai allydenedaspointswhere the ja obian matrix has rank

n − d

, and thus by the vanishing of some minors of the onsideredja obianmatrix. Ontheonehand,someofthese algorithmsallowusto obtain e ientimplementations(see[36℄) whilethealgorithmsmentionedintheaboveparagraph donotpermittoobtainusableimplementations. Ontheotherhand,applyingthe lassi al Bézoutboundtothepolynomialsystemsdeningthe riti allo usofaproje tionprovides adegreebound on theoutputwhi h isequalto

D

n−d

_{((n − d)(D − 1))}

d

(see[37, 7℄ where su haboundisexpli itlymentioned). Thisboundisworsethantheaforementionedbounds, butithasneverbeenrea hedin theexperimentsweperformedwithourimplementations.

Remark thatthesepolynomialsystemsdening riti alpointsarenotgeneri : theyare overdetermined,and the extra tedminors from theja obianmatrix depend on

f

1 , . . . , f

s

, sothat one anhopethat the lassi alBézoutboundis pessimisti .

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Ouraimistwofold:

providinganoptimalbound(inthesensethatit anberea hed)onthedegreeofthe riti allo usofapolynomialmappingrestri tedto analgebrai variety;

anddesigninganalgorithm omputingatleastonepointinea h onne ted omponent ofarealalgebrai setwhoseworst ase omplexityispolynomialin thisbound.

Main ontributions. Consideraproje tion

π

from

C

n

ontoalinewhosebase-lineve tor is

e

∈ C

n

, and the restri tion of

π

to the smooth algebrai variety

V ⊂ C

n

. Lagrange's hara terization of riti alpointsof

π

restri tedto

V

onsists in writingthat,ata riti al point, there exists alinear relation between the ve tors

(grad(f

1 ), . . . , grad(f

s

), e)

. The resultingpolynomialsystemis alledin thesequelLagrange's system. Additionalvariables areintrodu edand are lassi ally alled Lagrangemultipliers. Equipped withsu ha har-a terization, riti alpoints anbegeometri allyseenasproje tionsof the omplexsolution setof Lagrange's system. Weprovethat su ha hara terizationremainsvalid evenin non equidimensionalsituations, ontrarytothealgebrai hara terizationof riti alpointsused in [2, 35, 7,4, 5, 38, 37℄. If

f

1 , . . . , f

s

is a regular sequen e and if the riti al lo us of

π

restri tedto

V

iszero-dimensional,thenLagrange'ssystemisazero-dimensionalideal,else itisapositivedimensionalideal.

Sin e Lagrange multipliers appear with degree

1

in Lagrange's system, bounding the degree of the riti al lo us of

π

restri ted to

V

is equivalent to bounding the sum of the degrees of the isolated primary omponents of the ideal generated by Lagrange'ssystem. Lagrange'ssystem anbeeasilytransformedinto abi-homogeneouspolynomialsystemby abi-homogenizationpro esswhi hdistinguishthevariables

X

1 , . . . , X

n

fromtheLagrange multipliers. Thus,boundingthedegreeofa riti allo usisredu edtoprovingastrong bi-homogeneousBézoutTheorem,i.e. toprovingaboundonthesum ofthe degrees ofall the isolatedprimary omponentsdeninganon-emptybi-proje tivevarietyofanidealgenerated byabi-homogeneoussystem. Inthe sequel,thissumis alled thestrong bi-degreeofa bi-homogeneous ideal. Su h a result is obtained by using a onvenient notion of bi-degree, whi hisgivenoriginallyin[41℄. In[41℄,amulti-homogeneousBézouttheoremisprovedand providesaboundonthesumofthedegreesoftheisolatedprimary omponentsofmaximal dimension,whi hisnotsu ienttorea hourgoal.

Wegeneralizethisresultbyprovingthatthesamequantityboundsthesumofthedegrees of allthe isolated primary omponentsdening abi-proje tivevariety of an ideal dened byabi-homogeneouspolynomialsystem(see Theorem1). Additionally, weprovethatthe strongbi-degree of anideal generated byabi-homogeneous polynomialsystemequalsthe strongdegreeofthisidealaugmentedwithtwogeneri homogeneousanelinearformslying respe tivelyin ea hblo kofvariables(seeTheorem2).

Thisallowsustoprovethatthe riti allo usofaproje tion

π : C

n

_{→ C}

restri tedto

V

hasdegree

D

s

_{(D − 1)}

n−s

n

−s

. Thisboundbe omes

D

s

_{(D − 1)}

n−s n−1

n

−s

when

(f

1 , . . . , f

s

)

is a regular sequen e (see Theorem 3). Some omputer simulations show it is a sharp estimation,sin etheboundis rea hedonseveralexamples.

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Next,weusetheaforementionedpropertiesofLagrange'ssystemstogeneralize, tonon equidimensional situations,thealgorithm dueto Safey El Dinand S host(see [37℄)whi h omputesatleastonepointinea h onne ted omponentofasmoothandequidimensional realalgebrai set(seeTheorem5). Then,theestimationoftheoutputsizeofthisgeneralized algorithmprovidessomeimprovedupperboundsontherstBettinumberofasmoothreal algebrai set(seeTheorem6).

The omplexityofouralgorithmdependsonthe omplexityoftheroutineusedto per-form algebrai elimination. We onsider theelimination subroutine of [28℄ whi h inherits of[17, 18,19, 20℄. Thepro edureof [28℄ omnputesgeneri bersof ea h equidimensional omponentsofanalgebrai varietydened byapolynomialsystemprovidedasinput. Itis polynomialin theevaluation omplexityoftheinputsystem,andin anintrinsi geometri degree. This allows us to provethat the worst ase omplexity of ouralgorithm is poly-nomialin

n

,

s

,theevaluation omplexity

L

of

(f

1 , . . . , f

s

)

andthebi-homogeneousBézout bound

D

s

(D − 1)

n

−s

n

−s

n

(seeTheorem8).

Organizationofthepaper. Thepaperisorganizedasfollows. InSe tion2,weprovethe strongbi-homogeneousBézoutTheorem. Additionally,weprovethatthestrongbi-degreeof anidealequalsthestrongdegreeofthesameidealaugmentedwithtwogeneri anelinear formslyinginea hblo kofvariables. InSe tion3,wefo usonthepropertiesofLagrange's system and use our Bézout Theorem to prove somebounds on the degree of riti al lo i of proje tionson aline. In Se tion 4,wegeneralize the algorithm provided in [37℄ to the nonequidimensional ase. Moreover,usingtheresultsofthepre eding se tions,weprovide someimprovedupperboundsontherstBettinumberofasmoothrealalgebrai set. The lastse tionisdevotedtothe omplexityestimationofouralgorithm.

A knowledgments. WethankD.Lazard,J.Heintzandtheanonymousrefereesfortheir helpfulremarkswhi hhaveallowedustoimproveand orre tsomemistakesinthe prelim-inaryversionofthispaper.

2 Strong Bi-homogeneous Bézout theorem

This se tionis devoted to theproof of thestrong bi-homogeneousBézout Theorem. This onegeneralizesthestatementsof[40,41,31,32,24℄.

Intherstparagraph,weprovidesomeusefulpropertiesofbi-homogeneousidealswhi h allow to dene the notions of bi-degree and strong bi-degree of a bi-homogeneous ideal, thesenotionseemstogoba kto[40℄. Inthese ondparagraph,werelatethisbi-degreewith somepropertiesoftheHilbertbi-seriesofabi-homogeneousidealinthe asewhenitdenes azero-dimensional bi-proje tivevariety. Su h propertiesare already given in [40℄. Inthe thirdparagraph,weprovidea anoni alformofHilbertbi-series. Ourstatementsgeneralize slightlythoseof[31,32℄andallowustorelatetheHilbertbi-seriesofabi-homogeneousideal andtheHilbertseriesofthisideal,seenasahomogeneousone. Inthethirdparagraph,we investigatehowthebi-degreeofabi-homogeneousidealaugmentedwithabi-homogeneous

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polynomial isrelated tothe bi-degreeof therst onsideredideal. Finally, weprovethat, undersomeassumptions,the lassi albi-homogeneousBézoutboundisgreaterthanorequal to thesumof thebi-degrees of theprime idealsasso iatedto thestudied bi-homogeneous ideal. Thisgeneralizestheresultsof[40,31,32,24℄. Additionally,weprovethatthisquantity is greaterthan or equalto thesum of thedegreesof the primary idealsasso iatedto the studiedbi-homogeneousidealaugmentedwithtwoanelinearformswhi h generalizesthe results of [41, 32℄. For ompleteness and readibility, we give full proofs of the required intermediateresults.

Wedenoteby

R

thepolynomialring

Q[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

. 2.1 Preliminariesand main results

Inthisparagraph,weintrodu esomebasi sonbi-homogeneousideals. Weprovethatgivena bi-homogeneousideal

I ⊂ R

,thereexistsaminimalprimarybi-homogeneousde omposition of

I

,i.e. aprimary de ompositionsu hthat ea h primary omponentis abi-homogeneous ideal. Wedeneanotionofbi-degreeandstrongbi-degreeofbi-homogeneousideals. Finally, westatethemainresults(seeTheorems

1

and

2

below)ofthese tionwhi hboundthestrong bi-degreeofabi-homogeneousideal

I

bythe lassi albi-homogeneousBézoutboundonthe onehand,and showthat thestrong bi-degree of

I

bounds thedegreeof

I + hu − 1, v − 1i

(where

u

and

v

are homogeneous linear forms respe tively hosen in

Q[X

0 , . . . , X

n

]

and

Q[ℓ

0 , . . . , ℓ

k

]

).

Denition1 A linear formin

R

isa polynomialof degree

1

whose support ontains only monomialsof degree atmost

1

. A homogeneous linearformisalinear formwhose support ontainsonlymonomials ofdegree

1

.

A polynomial

f

in

R

is said to be bi-homogeneous if and only if there exists a unique oupleof integers

(α, β)

su hthatfor all

(u, v) ∈ Q × Q

:

f (uX

0 , . . . , uX

n

, vℓ

0 , . . . , vℓ

k

) = u

α

v

β

f (X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

n

).

The ouple

(α, β)

is alled thebi-degree of

f

.

Given a polynomial

f ∈ R

, the bi-homogeneous omponent of bi-degree

(α, β)

of

f

, de-noted by

f

α,β

, is the unique bi-homogeneous polynomial su h that the support of

f − f

α,β

does not ontainany monomialof bi-degree

(α, β)

.

An ideal

I ⊂ R

issaid tobeabi-homogeneousideal if andonlyiffor all

f ∈ I

,andfor allbi-degree

(α, β)

,

f

α,β

∈ I

.

Lemma1 Let

I ⊂ R

be a bi-homogeneous ideal. Then, there exists a nite polynomial family

f

1 , . . . , f

s

generating

I

su h thatea h

f

i

(for

i = 1, . . . , s

)isa bi-homogeneous poly-nomial.

Proof. Consideranitesetofgenerators

F

of

I

(thereexistsonesin e

R

isNoetherian). Sin e

I

isabi-homogeneousideal,theniteset

F

e

ofthebi-homogeneous omponentsofall thepolynomialsin

F

generatesanideal

J

whi his ontainedin

I

.

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Considernow

f ∈ I

. Sin e

I

isgeneratedby

F

andsin eea hpolynomialof

F

isalinear ombinationofsomepolynomialsin

F

e

,

I

is ontainedin

J

.

Lemma2 Let

Q

1 , . . . , Q

p

be afamily of bi-homogeneous ideals in

R

. Then

Q

1 ∩ · · · ∩ Q

p

isabi-homogeneousideal.

Proof. Denote by

I

theideal

Q

1 ∩ · · · ∩ Q

p

and onsider

f ∈ I

. Forall

i ∈ {1, . . . , p}

,

f ∈ Q

i

and

Q

i

is, by assumption, bi-homogeneous. Then, for all

i ∈ {1, . . . , p}

, ea h bi-homogeneous omponent of

f

belongs to

Q

i

, whi h implies that ea h bi-homogeneous omponentbelongsto

I

.

Givenabi-homogeneousideal

I ⊂ R

,wenowprovethat thereexists aprimary de om-positionof

I

forwhi hea hprimary omponentisbi-homogeneous.

Proposition 1 Let

I

be a bi-homogeneous ideal, and

Q

1 ∩ · · · ∩ Q

p

be aminimal primary de omposition of

I

. Then there exist primary bi-homogeneous ideals

Q

′

1 , . . . , Q

′

p

su h that

I = Q

′

1 ∩ · · · ∩ Q

′

p

.

Proof. For

i = 1, . . . , p

, onsider

Q

i

, a primary ideal of the above minimal primary de omposition of

I

and let

Q

′

i

be theideal generated by thebi-homogeneous polynomials

of

Q

i

. First, remarkthat

Q

′

i

is non empty sin eit ontains

I

. Weprovenowthat

Q

′

i

(for

i = 1, . . . , p

)isaprimaryidealandthenthat

I = Q

′

1 ∩ · · · ∩ Q

′

p

. Let

f

and

g

betwopolynomialssu hthat

f g ∈ Q

′

i

and

g /

∈ Q

′

i

. Weshowbelowthatthis

impliesthereexistsaninteger

N

su hthat

f

N

_{∈ Q}

′

i

whi h, inturn,implies

Q

′

i

isaprimary

ideal. Theproofisdonebyindu tiononthenumber

h

ofbi-homogeneous omponentsof

f

. If

h = 1

,

f

is bi-homogeneous, and the result is obvious. Suppose now that for any polynomial

f

e

having

h

bi-homogeneous omponents,

f g ∈ Q

e

′

i

with

g /

∈ Q

′

i

implies there

existsaninteger

N

su hthat

f

e

N

_{∈ Q}

′

i

.

Consider

f

having

h + 1

bi-homogeneous omponentsandsu hthat thereexists

g /

∈ Q

′

i

with

f g ∈ Q

′

i

. Sin e

Q

′

i

is bi-homogeneous,ea h bi-homogeneous omponentof

f g

belongs to

Q

′

i

. Remark that there exists one bi-homogeneous omponent of the produ t

f g

of maximal degreewhi h anbewritten as aprodu t

f

α,β

.g

α

′

,β

′

where

f

α,β

(resp.

g

α

′

,β

′

) is abi-homogeneous omponentof bi-degree

(α, β)

(resp.

(α

′

_{, β}

′

₎

) of

f

(resp.

g

). Moreover, withoutlossofgeneralityone ansuppose

g

α

′

,β

′

∈ Q

/

i

: ifitisnotthe ase,itissu ientto substitute

g

by

g − g

α

′

,β

′

.

Sin e

Q

i

is aprimaryideal, there exists

M ∈ N

, su hthat

f

M

α,β

∈ Q

i

. Moreover,sin e

f

α,β

is bi-homogeneous,

f

M

α,β

is bi-homogeneous. Thus, sin e

Q

′

i

is generated by the

bi-homogeneouspolynomialsof

Q

i

,thisimplies

f

M

α,β

∈ Q

′

i

. Sin e

f g ∈ Q

′

i

,

f

M

_{g ∈ Q}

′

i

. Moreover

f

M

_{g = (f − f}

α,β

)

M

g + f

α,β

M

g

while

f

M

α,β

∈ Q

′

i

. This implies

(f − f

α,β

)

M

_{g ∈ Q}

′

i

. Suppose now that

M

is the smallest integer su h that

(f − f

α,β

)

M

g ∈ Q

′

i

and remark that this implies

(f − f

α,β

)

M

−1

_{g /}

_{∈ Q}

′

i

. Thus, the above

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f

and

e

g = (f − f

α,β

)

M

−1

_{g /}

_{∈ Q}

′

i

insteadof

g

. Fromtheindu tion hypothesis, thisimplies there exists an integer

M

′

su h that

(f − f

α,β

)

M

′

∈ Q

′

i

, while theinteger

M

is su h that

f

M

α,β

∈ Q

′

i

. Thus, onsideringthebinomialdevelopmentof

(f −f

α,β

+ f

α,β

)

N

where

N > 2M

showsthat

f

N

_{∈ Q}

′

i

.

It remainsto provethat

I = Q

′

1 ∩ · · · ∩ Q

′

m

. Tothisend, remarkthat forall

i

,

I ⊂ Q

i

and as

I

is bi-homogeneous,

I ⊂ Q

′

i

while

Q

′

i

⊂ Q

i

. Thus one has

I ⊂ Q

′

1 ∩ · · · ∩ Q

′

m

⊂

Q

1 ∩ · · · ∩ Q

m

= I

whi h endstheproof.

Fromnowon,givenabi-homogeneousideal

I

,weonly onsiderbi-homogeneousprimary de ompositionsof

I

,i.e. de ompositionsof

I

su hthatea hprimary omponentis bi-homo-geneous. Remarkthatfromsu habi-homogeneousprimaryde omposition,one anextra t aminimalprimaryde omposition. Fromtheuniquenessoftheisolatedprimary omponents ofaminimal primary de omposition (see[12℄), onededu esthe uniquenessof theisolated primary omponentsofabi-homogeneousminimalprimary de ompositionof

I

. This leads tothefollowingresult.

Corollary1 Let

I

be a bi-homogeneous ideal of

R

. There exists a minimal primary de- ompositionof

I

su h that ea h primary omponent is a bi-homogeneous ideal. The set of isolatedbi-homogeneous omponents isunique.

Note that primary ideals of

R

whi h ontain a power of

hX

0 , . . . , X

n

i

or a power of

hℓ

0 , . . . , ℓ

k

i

deneanemptybi-proje tivevarietyin

P

n

_{(C) × P}

k

_(C)

.

Denition2 A primary bi-homogeneous ideal is said to be admissible if and only if it ontainsneitherapowerof

hX

0 , . . . , X

n

i

nor apowerof

hℓ

0 , . . . , ℓ

k

i

.

A primary omponent of a bi-homogeneous ideal is said tobe admissible if itis an ad-missibleideal.

The setofadmissible isolatedbi-homogeneous omponentsofaminimal bi-homogeneous primaryde ompositionof anideal

I ⊂ R

is denotedby

Adm(I)

.

Let

I ⊂ R

bea bi-homogeneousideal, and

(d, e)

bea ouple in

N

× N

. The ouple

(d, e)

is an admissible bi-dimension of

I

if and only if

d 6 n

,

e 6 k

and

d + e + 2

equals the maximumof theKrulldimensions ofthe ideals in

Adm(I)

.

Remark1 Consider a bi-homogeneous ideal

I ⊂ R

and let

J = ∩

Q∈Adm(I)

Q

. From Lemma2,

J

isabi-homogeneousideal.

A bi-homogeneousideal

I

in

R

denes anon-empty bi-proje tivevariety

V

in

P

n

_{(C) ×}

P

k

(C)

if and only if

Adm(I)

is not empty. Note also that the maximal dimension of the admissibleprimary omponentsof

I

anbelessthantheKrulldimensionof

I

.

Inthehomogeneous ontext,thedegreeofanidealofKrulldimension

D

deninga non-emptyproje tivevarietyisobtainedasthe degreeofthis idealaugmentedwith

D

generi linearforms (see[11℄). Below, weprovideasimilarpro essto dene anotionofbi-degree ofabi-homogeneousideal.

(12)

In the sequel,a homogeneouslinear form

u =

P

n

i=0

u

i

X

i

∈ Q[X

0 , . . . , X

n

]

(resp.

v =

P

k

j=0

v

j

ℓ

j

∈ Q[ℓ

0 , . . . , ℓ

k

]

)isidentiedtothepoint

(u

0 , . . . , u

n

) ∈ Q

n+1

(resp.

(v

0 , . . . , v

k

) ∈

Q

k+1

).

Proposition 2 Let

I ⊂ R

be an ideal of dimension

D > 2

and

(d, e) ∈ N × N

su h that

d + e + 2 = D

.

Either for any hoi e of homogeneous linear forms

(u

1 , . . . , u

d+1

)

in

Q[X

0 ,

. . . , X

n

]

and

(v

1 , . . . , v

e+1

)

in

Q[ℓ

0 , . . . , ℓ

k

]

the ideal

I + h(u

1 − 1), u

2 , . . . , u

d+1

, (v

1 − 1), v

2 , . . . , v

e+1

i

equals

Q[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

;

or there exist an integer

D

∈ N

and a Zariski losed subset

H ( (C

n+1

₎

d+1

_×

(C

k+1

₎

e+1

su hthatfor any hoi eofhomogeneouslinearforms

(u

1 , . . . , u

d+1

, v

1 , . . . ,

v

e+1

)

outside

H

(where for

i ∈ {1, . . . , d + 1}

,

u

i

∈ Q[X

0 , . . . , X

n

]

and for

j ∈

{1, . . . , e + 1}

,

v

j

∈ Q[ℓ

0 , . . . , ℓ

k

]

)the ideal

I + h(u

1 − 1), u

2 , . . . , u

d+1

, (v

1 − 1), v

2 , . . . , v

e+1

i

iszero-dimensionalanditsdegreeis

D

.

Proof. For

i ∈ {1, . . . , d + 1}

,

p ∈ {0, . . . , n}

,

j ∈ {1, . . . , e + 1}

, and

q ∈ {0, . . . , k}

, let

u

i,p

,and

v

j,q

benew indeterminates. Denote by

U

(resp.

V

) theset of indeterminates

{u

1,0

, . . . , u

d+1,n

}

(resp.

{v

1,0

, . . . , v

e+1,k

}

).

Let

K

betheeldofrationalfra tions

Q(U, V)

;for

i ∈ {1, . . . , d + 1}

,let

K

¯

i

betheeld ofrationalfra tions

Q

(U \ {u

i,0

, . . . , u

i,n

}, V)

andfor

j ∈ {1, . . . , e + 1}

,let

K

j

betheeld

ofrationalfra tions

Q

(U, V \ {v

j,0

, . . . , v

j,k

})

.

Considerfor

2 6 i 6 d + 1

(resp.

2 6 j 6 e + 1

)thelinearforms

u

i

=

P

n

p=0

u

i,p

X

p

(resp.

v

j

=

P

k

_q=0

v

j,q

ℓ

q

)and

u

1 =

P

k

p=0

u

1,p

X

p

− 1

(resp.

v

1 =

P

k

q=0

v

1,q

ℓ

q

− 1

).

For

i ∈ {1, . . . , d + 1}

(resp.

j ∈ {1, . . . , e + 1}

)and

u = (u

0 , . . . , u

n

)

(resp.

v =

(v

0 , . . . , v

k

)

) a point in

Q

n+1

(resp.

Q

k+1

), denote by

ϕ

i,u

(resp.

ψ

j,v

) the ring ho-momorphism

ϕ

i,u

: K[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

] → ¯

K

i

[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

(resp.

ψ

j,v

:

K[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

] → K

j

[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

)su hthat:

forall

p ∈ {0, . . . , n}

(resp.

q ∈ {0, . . . , k}

),

ϕ

i,u

(u

i,p

) = u

p

(resp.

ψ

j,v

(v

j,q

) = v

q

), for all

p ∈ {0, . . . , n}

(resp.

q ∈ {0, . . . , k}

), and

r ∈ {1, . . . , d + 1} \ {i}

(resp.

s ∈ {1, . . . , e + 1} \ {j}

),

ϕ

i,u

(u

r,p

) = u

r,p

(resp.

ψ

j,v

(v

s,q

) = v

s,q

),

ϕ

i,u

(X

p

) = X

p

(resp.

ψ

j,v

(X

p

) = X

p

)and

ϕ

i,u

(ℓ

q

) = ℓ

q

(resp.

ψ

j,v

(ℓ

q

) = ℓ

q

).

Finally, givena oupleof points

u = (u

1,0

, . . . , u

1,n

, . . . , u

d+1,0

, . . . , u

d+1,n

)

(resp.

v =

(v

1,0

, . . . , v

1,k

, . . . , v

e+1,0

, . . . , v

e+1,k

)

)in

(Q

n+1

₎

d+1

(resp.

(Q

k+1

₎

e+1

(13)

ringhomomorphism

ϑ

(u,v)

: K[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

] → Q[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

su h that forall

p = 0, . . . , n

and

q = 0, . . . , k

),

ϑ

u,v

(u

i,p

) = u

i,p

,

ϑ

u,v

(v

j,q

) = v

j,q

,

ϑ

u,v

(X

p

) = X

p

and

ϑ

u,v

(ℓ

q

) = ℓ

q

.

In the sequel,

J

0 = I

, for

i ∈ {1, . . . , d + 1}

,

J

i

is the ideal

I + hu

1 , . . . , u

i

, and for

i ∈ {d + 2, . . . , D}

,

J

i

istheideal

I + hu

1 , . . . , u

d+1

, v

1 , . . . , v

i

−(d+1)

i

.

We rst prove that the ideal

J

D

is either zero-dimensional in

K[X

0 , . . . , X

n

,

ℓ

0 , . . . , ℓ

k

]

or equals

K[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

. The proof is done by proving that, for

i = 1, . . . , D

,

dim(J

i

) < dim(J

i

−1

)

.

Consider the ase

i = 1

and suppose that

dim(J

1 ) > dim(J

0 ) > 0

. This implies that there exists an isolated primary omponent

Q

0

of

J

0

of dimension

dim(J

0 ) > 0

and an integer

N

su h that

u

N

1 ∈ Q

0

.

Sin e

J

0

isgeneratedbyapolynomialfamilywhi h doesnotinvolvetheindeterminates

u

1,0

, . . . , u

1,n

,foranyisolatedprimary omponent

Q

of

J

0

,

Q

is generatedbyapolynomial familywhi h does not involve the indeterminates

u

1,0

, . . . , u

1,n

. Thus, forany

u ∈ Q

n+1

,

ϕ

1,u

(Q

0 ) = Q

0 ∩ ¯

K

1

. Then,thereexistsaZariski- losedsubset

Z ⊂ C

n+1

su hthatforany point

u

in

Q

n+1

_{\ Z}

,

ϕ

1,u

(u

1 )

N

belongs to

Q

0

. Choose now

n + 2

points

u

1 , . . . , u

n+2

in

Q

n+2

su hthat

hϕ

1,u

1 (u

N

1 ), . . . , ϕ

1,u

n+2

(u

N

1 )i = h1i

. Sin e

hϕ

1,u

1 (u

N

1 ), . . . , ϕ

1,u

n+2

(u

N

1 )i ⊂ Q

0

,

Q

0 = K[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

whi h ontra-di ts

dim(Q

0 ) = D > 0

. Thus,

dim(J

1 ) < dim(J

0 )

.

Considernowthe asewhere

2 6 i 6 d + 1

and supposethat

dim(J

i

) > dim(J

i−1

) > 0

. Thisimpliesthatthereexistsanisolatedprimary omponent

Q

0

of

J

i

−1

ofpositivedimension andaninteger

N

su hthat

u

N

i

∈ Q

0

.

Sin e

J

i

−1

isgeneratedbyapolynomialfamilywhi hdoesnotinvolvetheindeterminates

u

i,0

, . . . , u

i,n

,

Q

0

is generated bya polynomialfamily whi h does notinvolve the indeter-minates

u

i,0

, . . . , u

i,n

and thus for any

u ∈ Q

n+1

,

ϕ

i,u

(Q

0 ) = Q

0 ∩ ¯

K

i

. Then, for any point

u

in

Q

n+1

,

ϕ

i,u

(u

i

)

N

belongs to

Q

0

. Choosing

n + 2

points

u

1 , . . . , u

n+2

in

Q

n+2

su h that

hϕ

i,u

1 (u

N

i

), . . . , ϕ

i,u

n+2

(u

N

i

)i = hX

0 N

, . . . , X

n

N

i ⊂ Q

. Sin e

u

1 ∈ J

i

−1

⊂ Q

0

and

hu

1 i + hX

0 N

, . . . , X

n

N

i = h1i

this impliesthat

Q

0 = K[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

whi h ontra-di tsthefa t that

Q

0

haspositivedimension. Thus,

dim(J

i

) < dim(J

i

−1

)

.

Provingthat

dim(J

d+2

) < dim(J

d+1

)

isdonebythesamewayasprovingthat

dim(J

1 ) <

dim(J

0 )

using the homomorphism

ψ

d+2,v

instead of

ϕ

1,u

(for

u ∈ Q

n+1

and

v ∈ Q

k+1

). Proving for

i > d + 2

that

dim(J

i

) < dim(J

i

−1

)

is done following the samearguments as thoseoftheaboveparagraph(usingthehomomorphisms

ψ

i,v

for

v ∈ Q

k+1

).

Thus,if

J

D

6= K[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

,onehas

dim(J

D

) < dim(J

D−1

) < · · · < dim(J

i

) <

dim(J

i−1

) < · · · < dim(I)

whi himplies

J

isbezero-dimensional. If

J

D

= K[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

, for any

(u, v) ∈ (Q

n+1

₎

d+1

_{× (Q}

k+1

₎

e+1

,

ϑ

u,v

(J

D

)

equals

Q[X

0 , . . . , X

n

]

.

Suppose now

J

D

to be zero-dimensional and onsider a Gröbner basis

G

of

J

D

. Let

H ⊂ (C

n+1

₎

d+1

_{× (C}

k+1

₎

e+1

betheZariki- losed setwhi h istheunionof zero-setsofthe ommondenominator of ea hpolynomialof

G

. Thus, for anypoint

(u, v) ∈ (Q

n+1

₎

d+1

_×

(Q

k+1

₎

e+1

_{\ H}

(14)

zero-dimensionalanditsdegreeis theoneof

J

D

.

Remark2 Fromthe proofof theabove Proposition, onededu esalso thefollowing:

Let

I ⊂ R

be a bi-homogeneous ideal whi h is an interse tion of admissible primary ideals. Then, there existsa Zariski- losed subset

H ( C

n+1

_{× C}

k+1

su h that for all

(u

1 , v

1 ) ∈ C

n+1

× C

k+1

\ H

,

u

1 − 1

and

v

1 − 1

do notdivide

0

in

R/I

and

v

1 − 1

does not divide

0

in

R/I + hu

1 − 1i

.

Moreoverif

I

isequidimensional,

I + hu

1 − 1, v

1 − 1i

isequidimensional.

Let

I ⊂ R

be an equidimensional bi-homogeneous ideal whi h is an interse tion of admissible primaryideals and

J = I + hu

1 − 1, v

1 − 1i

. If

J 6= R

,thereexists

(d, e) ∈

N

_{× N}

su hthat

d + e = dim(J)

andaZariski- losedsubset

H ( (C

n+1

₎

d

× (C

k+1

)

e+1

su h that for any hoi e of homogeneous linear forms

(u

2 , . . . , u

d+1

, v

2 , . . . , v

e+1

) ∈

Q[X

0 , . . . , X

n

] × Q[ℓ

0 , . . . , ℓ

k

] \ H

,

u

2

does not divide

0

in

R/J

,

u

i

does not divide

0

in

R/J + hu

2 , . . . , u

i

−1

i

(for

i = 3, . . . , d + 1

),

v

2

does not divide zero in

R/J +

hu

2 , . . . , u

d+1

i

and

v

j

does not divide

0

in

R/J + hu

2 , . . . , u

d+1

, v

1 , . . . , v

j−1

i

(for

j ∈

{3, . . . , e + 1}

).

Inthesequel,weshallsaythataproperty

P

istrueforageneri hoi eoflinearforms, ifthereexistsaZariski- losedsubsetin thesetofthe onsideredlinearformssu hthatfor any hoi eofforms outsidethisZariski- losedsubset,theproperty

P

issatised.

Proposition 2andtheuniqueness of

Adm(I)

allowsus todene thefollowingnotionof bi-degreeofabi-homogeneousideal

I ⊂ R

.

Denition3 Let

I ⊂ R

beabi-homogeneousidealofdimension

D

. For

(d, e) ∈ N × N

su h that

d + e + 2 = D

, onsiderthelinearforms

u

1 , . . . , u

d+1

(resp.

v

1 , . . . , v

e+1

)whi hare ho-sen generi ally in

Q[X

0 , . . . , X

n

]

(resp.

Q[ℓ

0 ,

. . . , ℓ

k

]

).

C

d,e

(I)

denotes the degreeof

I + hu

1 , . . . , u

d+1

− 1, v

1 , . . . , v

e+1

− 1i

. If

I

isprimary, the bi-degreeof

I

isthesum

P

d+e+2=D

C

d,e

(I)

.

If

I

isnotprimary,the bi-degreeof

I

isthesumofthebi-degreesofthe idealsof

Adm(I)

havingmaximalKrulldimension.

If

I

isnot primary,the strongbi-degreeof

I

isthesum ofthe bi-degreesofthe ideals of

Adm(I)

(whi h areisolatedbydenition of

Adm(I)

. Remark3 Let

I ⊂ R

beabi-homogeneousidealand

J = ∩

Q∈Adm(I)

Q

whi hisbi-homogeneous fromLemma2 . Notethat,bydenition,thebi-degree(resp. thestrongbi-degree)of

I

equals the bi-degree(resp. strongbi-degree) of

J

.

We are now ready to state the main results of this se tion. The rst onebounds the strongbi-degreeofabi-homogeneousideal

I ⊂ R

undersomeassumptions. Thisgeneralizes thestatementsof[40,39, 24℄ whi honly onsider withtheadmissible primary omponents of

Adm(I)

havingmaximaldimension.

(15)

Theorem1 Let

s ∈ {1, . . . , n + k}

and

f

1 , . . . , f

s

be bi-homogeneous polynomials in

R

of respe tive bi-degree

(α

i

, β

i

)

generating a bi-homogeneous ideal

I

. Suppose that there ex-ist at most

n f

i

su h that

β

i

= 0

and at most

k f

i

su h that

α

i

= 0

, then the sum of the bi-degrees of the bi-homogeneous asso iated primes of

I

is bounded by

B(f

1 , . . . , f

s

) =

X

I,J

(Π

i∈I

α

i

) . (Π

j∈J

β

j

)

where

I

and

J

aredisjointsubsetsforwhi htheunionis

{1, . . . , s}

su hthatthe ardinality of

I

(resp.

J

)isboundedby

n

(resp.

k

).

The following result generalizes the one of [41℄ and states that given an ideal

I ⊂

Q[X

1 , . . . , X

n

, ℓ

1 , . . . , ℓ

k

]

itsstrongdegree(inthemeaningof[23℄)isboundedbythestrong bi-degreeofabi-homogeneousideal onstru tedfrom abi-homogeneizationpro essapplied toea hpolynomialin

I

.

Theorem2 Consider themapping :

φ :

Q[X

1 , . . . , X

n

, ℓ

1 , . . . , ℓ

k

] → Q[X

0 , X

1 , . . . , X

n

, ℓ

0 , ℓ

1 , . . . , ℓ

k

]

f

7→ X

deg

X

(f )

0 ℓ

deg

_ℓ

(f )

0 f (

X

1

0 , . . . ,

X

n

X

0 ,

ℓ

1 ℓ

0 , . . . ,

ℓ

k

ℓ

0 )

where

deg

X

(f )

(resp.

deg

ℓ

(f )

)denotesthedegreeof

f

seenasapolynomialin

Q(ℓ

1 , . . . , ℓ

k

)[X

1 , . . . , X

n

]

(resp.

Q(X

1 , . . . , X

n

)[ℓ

1 , . . . , ℓ

k

]

).

Given an ideal

I ⊂ Q[X

1 , . . . , X

n

, ℓ

1 , . . . , ℓ

k

]

, denote by

φ(I)

the ideal

{φ(f) | f ∈ I} ⊂

Q[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

.

Then,

φ(I)

isabi-homogeneousidealandthe sum ofthe degreesof theisolatedprimary omponentsof

I

isboundedbythe strongbi-degreeof

φ(I)

.

TheproofoftheseresultsrelyonthestudyofHilbertbi-seriesofbi-homogeneousideals whi h areintrodu edin [40℄.

2.2 Hilbert bi-series: basi properties

We follow here [40℄ whi h introdu e Hilbert bi-series of bi-homogeneous ideals and study theirpropertieswhenthe onsideredbi-homogeneousidealhasbi-dimension

(0, 0)

.

Notation. Given a ouple

(i, j) ∈ N × N

, we denote by

R

i,j

the

Q

-ve tor spa e of the bi-homogeneouspolynomialsin

R

ofdegree

i

inthesetofvariables

X

0 , . . . , X

n

and

j

inthe setofvariables

ℓ

0 , . . . , ℓ

k

.

I ⊂ R

and a ouple

(i, j) ∈ N × N

,wedenoteby

I

i,j

the interse tionof

I

with

R

i,j

.

Givena ouple

(i, j) ∈ N × N

, wedenote by

R

6i,6j

the

Q

-ve torspa eof polynomials in

R

of degree less than or equal to

i

(resp.

j

) in the set of variables

X

0 , . . . , X

n

(resp.

ℓ

0 , . . . , ℓ

k

).

Givenanideal

I ⊂ R

anda ouple

(i, j) ∈ N × N

,wedenote by

I

6i,6j

theinterse tion of

I

with

R

6i,6j

.

(16)

Denition4 TheHilbertbi-seriesofabi-homogeneousideal

I ⊂ R

istheseries

P

i,j

dim (R

i,j

/I

i,j

) t

i

1 t

j

2

. The aneHilbertbi-series of anideal

I ⊂ R

is

P

i,j

dim (R

6i,6j

/I

6i,6j

) t

i

1 t

j

2

.

Considerabi-homogeneousideal

I ⊂ R

ofKrulldimension

2

. Thisse tionisdevotedto provethat,thereexists

(i

0 , j

0 ) ∈ N × N

su hthatforall

i > i

0

and

j > j

0

,

dim(R

i,j

/I

i,j

) =

dim(R

i

0 ,j

0 /I

i

0 ,j

0 )

. Su haresultalreadyappearsin[40℄. Denote by

R

′

the polynomial ring

Q[X

1 , . . . , X

n

, ℓ

1 , . . . , ℓ

k

]

. Consider the appli ation

φ

i,j

: R

′

6i,6j

→ R

i,j

sendingapolynomial

f ∈ R

′

6i,6j

ofdegree

α

(resp.

β

)inthevariables

X

1 , . . . , X

n

(resp.

ℓ

1 , . . . , ℓ

k

)to thepolynomial

φ

i,j

(f ) = X

i

0 ℓ

j

0 f (

X

1

0 , . . . ,

X

n

X

0 ,

ℓ

1 ℓ

0 , . . . ,

ℓ

k

ℓ

0 )

. Givenanideal

I ⊂ R

′

and

(i, j) ∈ N × N

,

φ

i,j

(I)

denotestheset

{φ

i,j

(f ) | f ∈ I

6i,6j

}

. Additionally, onsider the mapping

ψ

i,j

: R

i,j

→ R

′

6i,6j

(whi h takes pla e of a

de-homogenization pro ess in a homogeneous ontext) sending abi-homogeneous polynomial

f ∈ R

i,j

to

ψ

i,j

(f ) = f (1, X

1 , . . . , X

n

, 1, ℓ

1 , . . . , ℓ

k

) ∈ R

′

.

I ⊂ R

and

(i, j) ∈ N × N

,

ψ

i,j

(I)

denotestheset

{ψ

i,j

(f ) |

f ∈ I

i,j

}

.

Lemma3 Consider twointegers

i

and

j

, an ideal

I

′

of

R

′

and

φ

i,j

the above appli ation from

R

′

6i,6j

to

R

i,j

. Then

dim(R

′

6i,6j

/I

6i,6j

′

) = dim(R

i,j

/φ

i,j

(I

6i,6j

′

)).

Proof. Remark that

R

i,j

,

R

′

6i,6j

, and

I

′

6i,6j

and

φ

i,j

(I

′

6i,6j

)

are nite dimensional

Q

-ve torspa es.

Moreover,for all

f ∈ R

′

6i,6j

and

(i, j) ∈ N × N

,

ψ

i,j

(φ

i,j

(f )) = f

. Then,

ψ

i,j

(R

i,j

) =

R

′

6i,6j

and

ψ

i,j

(φ

i,j

(I

′

6i,6j

)) = I

6i,6j

′

. Hen e as

ψ

i,j

is an inje tive morphism,

R

i,j

and

R

′

6i,6j

ontheonehand,and

I

′

6i,6j

and

φ

i,j

(I

′

6i,6j

)

ontheotherhand,areisomorphi nite dimensional

Q

-ve torspa es.

Sin eforve torspa es

E, F

with

F ⊂ E

,

dim(E) = dim(F ) + dim(E/F )

,wearedone.

Thefollowinglemmaisusedfurther.

Lemma4 Let

I ⊂ R

beabi-homogeneousidealsu hthat

X

0

(resp.

ℓ

0

)isnotazerodivisor in

R/I

. Denote by

I

′

the ideal

I + hX

0 − 1, ℓ

0 − 1i ∩ R

′

. Then

φ

i,j

(I

′

6i,6j

) = I

i,j

. Proof. Considera bi-homogeneous polynomial

p ∈ I

i,j

. Obviously,

ψ

i,j

(p) ∈ I

′

6i,6j

.

Sin e the bi-degree of

p

is, by denition, the ouple

(i, j)

,

φ

i,j

(ψ

i,j

(p)) = p

, this implies

p ∈ φ

i,j

(I

6i,6j

′

)

. Thus,

I

i,j

⊂ φ

i,j

(I

′

6i,6j

)

;weprovenowthat

φ

i,j

(I

′

6i,6j

) ⊂ I

i,j

.

FromLemma1,sin e

I

isbi-homogeneous,thereexistsanitefamilyofbi-homogeneous polynomials

p

1 , . . . , p

m

generating

I

. Consider

p

′

_{∈ I}

′

6i,6j

. Then, there exist polynomials

q

r

(for

r ∈ {1, . . . , m}

),

P

and

Q

in

R

,su hthat

p

′

₌

P

m

r=1

q

r

.p

r

+ Q(X

0 − 1) + P (ℓ

0 − 1)

. Denoteby

p

thepolynomial

P

m

(17)

allthebi-homogeneous omponentsof

p

belongto

I

,andonejust hasto onsider the ase where

p

′

issu hthat

p

isbi-homogenous.

Under this assumption, it is easy to see that

φ

i,j

(p

′

₎

belongs to

R

i,j

and that there existsa ouple

(α, β) ∈ Z × Z

su hthat

φ

i,j

(p

′

_{) = X}

α

0 .ℓ

β

0 .p

. If both

α, β

arepositive,then

φ

i,j

(p

′

) ∈ I

i,j

. If

α < 0

and

β > 0

, then

X

−α

0 φ

i,j

(p

′

) = ℓ

β

0 p ∈ I

. Suppose

φ

i,j

(p

′

_{) /}

_{∈ I}

. This would imply that

X

0

is a zero-divisorin

R/I

, whi h is notpossible by assumption. Moreover,sin e

φ

i,j

(p

′

₎

hasbi-degree

(i, j)

,

φ

i,j

(p

′

₎

belongsto

I

i,j

. Similarargumentsallow usto on ludewhen

α > 0

and

β < 0

andwhenboth

α

and

β

arenegative.

Givenapolynomial

f ∈ Q[X

0 , . . . , X

n

, ℓ

0 , . . . , ℓ

k

]

, and

A

∈ GL

n+k+2

(Q)

, wedenoteby

f

A

thepolynomialobtainedbyperformingthe hangeofvariablesindu edby

A

on

f

. We denote by

I

A

the ideal generated by

f

A

1 , . . . , f

s

A

. In the sequel, we onsider ex lusively matri es

A

su h that the a tionof

A

on

R

is abi-gradedisomorphismof bi-degree

(0, 0)

on

R

withrespe ttothevariables

X

0 , . . . , X

n

and

ℓ

0 , . . . , ℓ

k

,i.e. forallhomogeneouslinear forms

u ∈ Q[X

0 , . . . , X

n

]

(resp.

v ∈ Q[ℓ

0 , . . . , ℓ

k

]

),

u

A

(resp.

v

A

) isa homogeneouslinear formin

Q[X

0 , . . . , X

n

]

(resp.

Q[ℓ

0 , . . . , ℓ

k

]

).

Lemma5 Let

I ⊂ R

be a bi-homogeneous ideal, then

I

and

I

A

have the same Hilbert bi-series.

Proof. Thea tion of

A

on

R

is anisomorphismof bi-graded ringof bi-degree

(0, 0)

, the inverse a tion of

A

is the a tion of

A

−1

. Thus,

R

i,j

equals

R

A

i,j

and, if

E ⊂ R

is a

Q

-ve torspa e,

dim(E) = dim(E

A

₎

. Sin e forve torspa es

E, F

with

F ⊂ E

,

dim(E) =

dim(F ) + dim(E/F )

,wehave

dim(R

i,j

/I

i,j

) = dim(R

A

i,j

/I

i,j

A

)

whi himpliestheequalityof theHilbertbi-seriesof

I

and

I

A

.

Lemma6 Let

I ⊂ R

beabi-homogeneousidealwhi hisaninterse tionofadmissibleideals. Then,forageneri hoi eofhomogeneouslinearform

u

1

(resp.

v

1

)in

Q[X

0 , . . . , X

n

]

(resp.

Q[ℓ

0 , . . . , ℓ

k

]

) theaneHilbertbi-series of

I + hu

1 − 1, v

1 − 1i

equals theHilbertbi-series of

I

.

Proof. FromRemark2,sin eis

u

1

and

v

1

are hosengeneri allytheydonotdivide

0

is

R/I

. Choose

A

∈ GL

n+k+2

(Q)

su h thatthea tionof

A

on

R

isabi-gradedisomorphism ofbi-degree

(0, 0)

andsu hthat

u

A

1 = X

0

and

v

A

1 = ℓ

0

. UsingLemma 5,theHilbert series of

I

equals the oneof

I

A

. From Lemma 3 and Lemma 4, the ane Hilbert bi-series of

I

A

_{+ hX}

0 − 1, ℓ

0 − 1i ∩ Q[X

1 , . . . , X

n

, ℓ

1 , . . . , ℓ

k

]

equalsthebi-seriesof

I

A

whi hequalsthe oneof

I

.

We prove now

ψ

′

i,j

: R

A

6i,6j

→ R

′A

6i,6j

sending

f ∈ R

A

6i,6j

to

f (1, X

1 , . . . , X

n

,

1, ℓ

1 , . . . , ℓ

k

)

indu es an isomorphism between

R

′A

6i,6j

/I

′A

6i,6j

and

R

A

6i,6j

/(I

A

+ hX

0 − 1, ℓ

0 − 1i

6i,6j

)

whi hisimmediateusingtheisomorphismbetween

R

′A

6i,6j

/I

′A

6i,6j

and

R

A

i,j

/I

i,j

A

.

(18)

ThefollowingresultisusedintheproofofTheorem2.

Proposition 3 Let

I

be abi-homogeneous ideal of

R

. Then, the Hilbert seriesof

I

equals the seriesobtainedbyputting

t

1 = t

2

inthe Hilbert bi-series of

I

.

Proof. For all

d ∈ N

,denoteby

R

d

the

Q

-ve torspa eofthehomogeneouspolynomials of

R

of totaldegree

d

. Forany ouple

(i, j) ∈ N × N

, denote by

R

i,j

the

Q

-ve torspa e ofbi-homogeneouspolynomialsofbi-degree

(i, j)

. Bydenition ofabi-homogeneousideal, if

f

is a polynomial of

I

, then the bi-homogeneous omponents of

f

also belong to

I

. Hen e themorphism from

I

d

to

Π

i+j=d

I

i,j

sending apolynomialonto itsbi-homogeneous omponentsis dened for any

f

of

I

and is invertible, i.e. is an isomorphismof

Q

-ve tor spa es. Thisprovesthat

dim(I

d

) =

P

i+j=d

dim(I

i,j

)

. With thesamearguments,weshow that

dim(R

d

) =

P

i+j=d

dim(R

i,j

)

. Consequently

dim(R

d

/I

d

) = dim(R

d

) − dim(I

d

) =

X

i+j=d

dim(R

i,j

) − dim(I

i,j

)

Ontheother hand,forall

(i, j) ∈ N × N

,

dim(R

i,j

) − dim(I

i,j

) = dim(R

i,j

/I

i,j

)

.

Proposition 4 Let

I ⊂ R

be a bi-homogeneous ideal and let

J = ∩

Q∈Adm(I)

Q

. Suppose that

J

has Krull dimension

2

. There exists

(i

0 , j

0 ) ∈ N × N

su h that for all

i > i

0

and

j > j

0

,

dim(R

i,j

/J

i,j

) = dim(R

i

0 ,j

0 /J

i

0 ,j

0 )

whi h equalsthe bi-degree of

I

.

Proof. FromRemark3,thebi-degreeof

I

isthebi-degreeoftheideal

J = ∩

Q∈Adm(I)

Q

. FromProposition2andDenition3,thebi-degreeof

J

isthedegreeoftheideal

J + hu −

1, v − 1i

where

u

(resp.

v

) is a generi homogeneouslinear form in

Q[X

0 , . . . , X

n

]

(resp.

Q[ℓ

0 , . . . , ℓ

k

]

).

Consider now

A

∈ GL

n+k+2

(Q)

su h that

u

A

_{= X}

0

and

v

A

_{= ℓ}

0

, and su h that the

anoni ala tionof

A

on

R

isabi-gradedisomorphismon

R

ofbi-degree

(0, 0)

withrespe t to thevariables

X

0 , . . . , X

n

and

ℓ

0 , . . . , ℓ

k

. Notethat the bi-degree of

J

equalstheoneof

J

A

whi h is thedegree of

J

A

_{+ hX}

0 − 1, ℓ

0 − 1i

. Inthe sequel, denote by

J

′A

theideal

J

A

_{+ hX}

0 − 1, ℓ

0 − 1i

∩ Q[X

1 , . . . , X

n

, ℓ

1 , . . . , ℓ

k

]

From Lemma 5,

dim(R

i,j

/J

i,j

) = dim(R

A

i,j

/J

i,j

A

)

. Moreover, sin e by denition of

J

A

,

X

0

(resp.

ℓ

0

) is not a zero-divisor in

R

A

_/J

A

, from Lemma 4,

dim(R

A

i,j

/J

i,j

A

) =

dim(R

′A

i,j

/φ

i,j

(J

′A

6i,6j

))

. Finally,fromLemma3

dim(R

′A

i,j

/φ

i,j

(J

′A

6i,6j

)) = dim(R

′A

6i,6j

/J

′A

6i,6j

)

. Thus, one has

dim(R

i,j

/J

i,j

) = dim(R

′A

6i,6j

/J

′A

6i,6j

)

. It is su ient to prove there ex-ists

(i

0 , j

0 ) ∈ N × N

su h that for all

(i, j) ∈ N × N

satisfying

i > i

0

and

j > j

0

,

dim(R

′A

6i,6j

/J

′A

6i,6j

)

= dim(R

′A

6i

0 ,6j

0 /J

′A

6i

0 ,6j

0 )

and that

dim(R

′A

6i

0 ,6j

0 /J

′A

6i

0 ,6j

0 )

equals the degree of

J

′A

.

Thesearea onsequen eof

dim(J

′A

_{) = 0}

andfor

i

,

j

and

d

largeenough

dim(R

′A

6i,6j

/J

′A

6i,6j

) =

dim(R

′A

6d

/J

′A

6d

) = deg(J

′A

)

,where

R

′A

6d

(resp.

J

′A

6d

)denotesthesetofpolynomialsin

R

′A

(19)

(resp.

J

′A

)ofdegreelessthanorequalto

d

.

2.3 Canoni al form of the Hilbert bi-series

Inthisparagraph,weprovidea anoni alformoftheHilbert bi-seriesofabi-homogeneous ideal

I ⊂ R

with respe t to the integers

C

d,e

(I)

where

(d, e)

lies in the set of admissible bi-dimensionsof

I

.

Lemma7 Let

I ⊂ R

beabi-homogeneousideal. For

i = 1, . . . , s

let

f

i

beabi-homogeneous polynomialin

R

of bi-degree

(α

i

, β

i

)

. For

i = 1, . . . , s

,denoteby

I

i

the ideal

I + hf

1 , . . . , f

i

andsupposethat for all

i ∈ {1, . . . , s − 1}

,

f

i+1

isnotadivisor of zeroin

R/I

i

.

Then the Hilbertbi-series of

I

s

equals

Π

s

i=1

(1 − t

α

i

1 t

β

i

2 )

H(I)

.

Proof. Wepro eedbyindu tionon

s

. Supposerst

s = 1

.

Denoteby

ann

R/I

(f

1 )

theannihilatorof

f

1

in

R/I

. Thesequen ebelow

0 → ann

R/I

(f

1 ) −→ R/I

f

1 −→R/I −→ R/(I + hf

1 i) → 0

is exa t. Sin e

f

1

is nota zerodivisor in

R/I

,

ann

R/I

(f

1 )

is

0

. Remark that, sin e

f

1

is bi-homogeneous,and

I

i,j

and

R

i,j

ontainonlybi-homogeneouspolynomials,thefollowing one

0 −→ R

i,j

/I

i,j

f

1 −→R

i+α

1 ,j+β

1 /I

i+α

1 ,j+β

1 −→ R

i,j

/(I + hf

1 i)

i,j

→ 0

is alsoexa t. Thus, lassi ally,the alternatesum ofthe dimensionof theve torspa esof thisexa tsequen eisnull. Addingthesesumsforall