Strong Bi-homogeneous Bézout's Theorem and degree bounds for algebraic optimization

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Submitted on 23 May 2006

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Strong Bi-homogeneous Bézout’s Theorem and degree bounds for algebraic optimization

Mohab Safey El Din, Philippe Trebuchet

To cite this version:

Mohab Safey El Din, Philippe Trebuchet. Strong Bi-homogeneous Bézout’s Theorem and degree

bounds for algebraic optimization. [Research Report] RR-5071, INRIA. 2004. �inria-00071512�

ISRN INRIA/RR--5071--FR+ENG

a p p o r t

d e r e c h e r c h e

THÈME 2

INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Strong Bi-homogeneous Bézout’s Theorem and degree bounds for algebraic optimization

Mohab Safey El Din — Philippe Trébuchet

N° 5071

Janvier 2004

Unité de recherche INRIA Rocquencourt

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mozw~Dm¨ÂËrs„®w_ne[)mozq_dÃqs„}ŽDbDmzd qsl

R

^¬ mu®b„}e[z}Ž§e1t1uu®q¶u®bneKš_w.„m©w.~Dpoe1z

X 1 , . . . , X n

w_ln

` 1 , . . . , ` k

^±

m ¸e ¸

ªÉq_„w_pop§bDqsdfqsrse1lDe[qsynzpomolnexw_„

ªÉqs„}dÃz

X ∈ Q[X 1 , . . . , X n ]

^{« „e1zŽ ¸}

L ∈ Q[` 1 , . . . , ` k ]

^{¯ ±}

A(X)

^{« „e[z}Ž ¸}

A(L)

^¯ mzpomlDexw.„ªÉqs„}d mol

Q[X 1 , . . . , X n ]

^{« „}e[zŽ ¸}

Q[` 1 , . . . , ` k ]

^{¯ ¸}

…Ø ¢ Bº

I ⊂ R

^{7º (_»+½! \º (*) *º7»}

I

^(_»

I ^A

^{%(_º º+$ (*"Àº $‰½#) 7º7¼} ! [½"'$¡º7¼ ½¿º $ Ƽ &&

abDew_t1umoqsl)q_ª

A

^qsl

R

mozvw_l´mzqsdfqs„}ŽDbDmoz}dq_ª&~Dm¨ÂËrs„®w_ne[)„molnrÃq_ª&De[r_„e[e

(0, 0)

^{ªÉqs„ ¬ bDmotb}

u®bne¶molGšse1„ze¶mz‰u®bnew_t1umoqslXq.ª

A ⁻¹

^{¸ ab‚yDz ±}

R i,j

e1€‚yw_poz

R _i,j ^A

^{w.lD ± m¨ª}

E ⊂ R

^mozkw

Q

¿šse[t7u®qs„kz}Žw_t1e

±

dim(V ) = dim(E ^A )

^¸ ˆ˜mlDt[eͪÉqs„šse1t1u®q_„²z}Žw_t[e1z

E, F

^{¬ mub}

F ⊂ E

^±

dim(E) = dim(F ) +

dim(E/F )

^{±n¬ ebwxš_e}

dim(R i,j /I i,j ) = dim(R i,j /I _i,j ^A )

^¬ bDmotb²modfŽDpome[zSu®bDe%e[€‚yw.pomu·|q_ªubDe

mop~§e1„}u

~Dmηze1„me[zq_ª

I

^w_ln

I ^A

^¸

  Ü Ü

žA­Ÿ

Ü 6º

I ⊂ R

^{7º (_»+½! \º} (*)?¹\º7»<º7¼4(·º [½" &3" &¹\º7»<º & $ &3)‡» &3"¶½!(*)$ \º,+…»½­»\¹ ( ‡º7¼ & " _½#"Àº7»%$7½'&‡»(*) [½" ?¼ &xº¾T½_º3(_¼ ½¿º

V

^½­»

P ⁿ⁻¹ (C) × P ^k−1 (C)

^-1&‡¼

i

^w_lD

j

p©w_„}rsee[lDq_yDrsb º¶¾ &¡º,& ¾7½¿º7» &k½­» \º

i, j

^{½­» º $K½#) 7º7¼} ! [½"'$¡º7¼7½¿º $ &

I

dim(R i,j /I i,j )

^{½$¾ &‡»%$(_»}

#$ 3(*)“º½ $º \º¿¹_¼®ºº& º¶½! \º (*)

I + hX − 1i + hL − 1i ⊂ R

^º7¼®º

X

^(_»

L

(_¼®º%¹\º7»<º7¼ ½¿¾

&3" &¹\º7»<º & $+)½­»<º (_¼ A&‡¼?" $¾ &$¡º7»X¼®º $ º¾T½_º )f½­»

Q[X 1 , . . . , X n ]

^(_»

Q[` 1 , . . . , ` k ]

Ƽ &&

i%m¨šse1l+rse[lDe1„mtbDqsdfqsrse1lDe[qsynzkpomolnexw_„ªÉqs„dfz

X ∈ Q[X ₁ , . . . , X _n ]

^w_lD

L ∈ Q[` <sub>1</sub> , . . . , ` _k ]

z®w‡u®mozŪ­|\molDru®bDew_zz}yDdfŽnu®mqslDzBq_ª ¦ e[dfdÀwMg

±

u®bDeÆmoDe[w_p

I +hX −1i+hL−1i ⊂ R

mzÁ1e[„q‡Â·DmdÃe1lDzmqslw_p w_lDÍm¨u®zkDe1rs„e1efnq˜e1zlnq_u%De[Ž?e[ln qsl+u®bne¶tbDqsmot1eq_ª

X

^w_ln

L

^{¸ q ¬%±} tbDq\qszmlDr

A ∈ GL n+k ( Q )

zyntbµu®bDw.u

A(X) = X _n

^w.lD

A(L) = ` _k

^± w_lDÑzyntbÑubw.uku®bneft[w_lDqslnmotxw.pVw_t1umoqsl+q.ª

A

^qsl

R

^moz%w

~Dmηrs„w_De[Ñmoz}qsdfqs„ŽDbnmozd qsl

R

q.ªDe[r_„e[e

(0, 0)

^¬ mubµ„}e[z}Ž§e1t1uuq²u®bnefš.w_„}m©w_~Dpe[z

X 1 , . . . , X n

w.lD

` 1 , . . . , ` k

^±

u®bDe%t[q_lDt[pyDzmqslªÉqspopq

¬

zmodfdÃe1Dm©w.ue[p¨|fªÉ„}qsd ¦ e1dÃdÃw.uw›fw_ln ¸

Ç

„qsd‘ubDeKŽD„}e[t[e1e[DmlDrŽw_„w_rs„®w.ŽDb

±

u®bDe (*)¹\º [¼4(_½¿¾ [½" \º¿¹_¼®ººSq_ªBw~DmηbDqsdfqsrse1lDe[q_yDzmDexw_p?De7³lDmolnr¶w

~DmηŽD„}q‡†}e[t7u®mš_evš.w_„}moe7u·|Ãml

P ⁿ⁻¹ (C) × P ^k−1 (C)

txw.l~§eMDe1³DlDe[w_zSu®bDeMt[pow_zz}motxw.p?ne[rs„}e[eKq_ª<u®bDeMz®w_dfe monexw_pw_yDrsdfe[lGue[X~G|)u

¬

qw/flDepomolnexw_„ªÉqs„dfz

±

u®bne³„zÅuKqslnep|\molnrÀmol

Q[X 1 , . . . , X n ]

qsl´ubDeqslne bw_ln

±

w.lDÃubDeMq.u®bDe1„Æqslne‰p|\mlDr%mol

Q[` <sub>1</sub> , . . . , ` _k ]

qslfu®bDevq_u®bne[„Sbw_lD

¸

e[dÃw_„4,u®bw‡uÆubDmozÆw_por_e[~D„w_mot

~DmηDe1rs„e1emozKw_poz}qÃu®bnezyDdÐq_ªVu®bne¶w_por_e[~D„w_mot~Dm¨ÂˏDe[r_„e[e1zKq_ªVubDew_Ddfmoz}zmo~npoe%ŽD„mdÀw.„}|)t[qsdfŽ?qslDe[lGuz

q_ª!ubDeÀt1qslDz}moDe1„e[ moDe[w_p

¸

aBqXDe7³lDew²lDq_u®mqslÑq_ªw_prse[~D„w_mot~DmηDe1rs„e1eÀwxš.w_mp©w_~Dpe¶molÑu®bDeΧqsz}mumšse

DmdÃe1lDzmqslw_p6txw.ze

±{¬

e¶yDz}eu®bDe¶Žn„e[t1e[e1DmolDr~G|Xw_yDrsdfe[lGu®mlDrÃubDe¶monexw_pByDlDDe1„kt[qslnzmone[„®w‡u®moq_l

¬

mub

bDqsdfqsrse1lDe[q_yDz!pomlDexw.„ƪÉqs„}dÃzÆmol

Q[X 1 , . . . , X n ]

^w_ln

Q[` 1 , . . . , ` k ]

„}e[Dynt[molnr¶qsyD„!zÅu®yDn|Àu®qubDektxw_z}e q_ª‰~Dm¨ÂËbDqsdfqsrse[lne[qsyDz¶mDexw.pozq_ªK~nm¨Â·nmodfe[lDz}moqsl

(0, 0)

^¸ Š‰zfdÃe1l‚umoqslnlDe[ w_~?q£š_e

±

ubDmozt[w_l ~?e²nqslDe

mol ze7šse[„w_p

¬

wx|\z

¸

abDe²„e1zyDp¨u~?e[poq

¬

mzÀt1„yDt1m©w_pu®qµyDlDDe1„zÅuw_lD ubDe´t[w_lDqslnmot[m¨u·| q_ªku®bne²ªÉyD„Åu®bDe1„

De7³lDmumoqslnzq_ª2w.porse1~D„®w_mtK~DmηDe1rs„e1ekrsmš_e[l)molu®bnmozŽw_„w_rs„®w.ŽDb

¸

…Ø ¢ Bº

I ⊂ R

^{7º ( [½"} &3" &¹\º7»<º & $K½! \º (*)9(_» $ &$¡º º "( s½#"(*)9 _½#"Àº7»%$7½'&‡» & *º ( *"¶½$ $7½' )oº ?¼ ½#"(_¼ Ѿ &3" &‡»<º7»#$&

I

^{& 7º}

3

&‡»%$7½! \º7¼

f ∈ R

^{& [½" \º¿¹_¼ºº}

(α, β )

^\½¿¾Ñ½$

¹\º7»<º7¼ ½¿¾ A&‡¼

I

^{(˽ º}

f

^{&¡º $k» & 7º )&‡»\¹ &7(_» ?¼ ½#"Àº (3$} $&¡¾7½!("º &( ?¼7½#"(_¼ À¾ &3" &‡»<º7» &

I

^*

º7»A&‡¼¹\º7»<º7¼ ½¿¾ )½­»<º (_¼ &3" &¹\º7»<º & $ A&‡¼?" $

u, X

^½­»

Q[X 1 , . . . , X n ]

^(_»

v, L

^½­»

Q[` 1 , . . . , ` k ]

deg(I +hf i+hX−1i+hL−1i) = α deg(I +u+hX −1i+hL−1i)+β deg(I +v+hX −1i+hL−1i).

Ƽ &&

¦

e1u

H(I) = P

i,j a i,j t ⁱ ₁ t ^j ₂

De1lDq_u®e%u®bne

mopo~?e[„Åu~Dm¨ÂËze1„moe1zq_ª

I

^±

ann R/I (f )

ubDew.lDlDmobnmop©w‡u®qs„

q_ª

f

^¬ m¨u®b+„}e[zŽ?e[t7uKu®q

I

^¸ ve1dÀw_„4,u®bw.uKzmolnt[e

f

mz‰rse1lDe[„}motªÉq_„

I

^±

ann R/I (f )

^moz

0

w_lD´ubw.uKzmolnt[e

dim(I + hf i + hX − 1i + hL − 1i) = 0

^¸ ˆ˜mlDt[eKubDe%ze[€‚yDe1lDt[e~§e1poq

¬

0 → ann R/I (f) −→ R/I −→R/I ^f −→ R/(I + hf i) → R

moze Ìnw_t1u

±

ubDeKªÉqsppoq

¬

molDrbDqsponz!ªÉqs„w_lG|rse[lne[„mtke[pe[dfe[lGu

f ∈ R

q_ª~DmηDe1rs„e1e

(α, β)

⁵

(1 − t ^α ₁ t ^β ₂ )H(I) = H(I + hfi)

abDeKzw_dfe

¬

w¡|

±

„}e[dÃw_„ ,fw_pzqubw.uƪÉqs„wrse[lDe1„mt‰pomlDexw.„…ªÉq_„d

u

^{« „}e[zŽ ¸}

v

^{¯ mol}

Q[X 1 , . . . , X n ]

^{«„}e[zŽ ¸}

Q[` 1 , . . . , ` k ]

^¯ ubDe~DmηDmdÃe1lDzmqslq.ª

I +hui

^{«„e[z}Ž ¸}

I +hvi

^{¯ moz}

(0, 0)

w.lDu®bDe1l¶u®bDe

mop~§e1„}u2~nm¨Â·z}e[„}moe[z

H(I + hui)

^w_lD

H(I + hv i)

bwxšsew¶t[qslnz}uw.l‚uue[„d

¬

bDe[l²~§q_ub

i, j → ∞

^{± zwx|}

S 1

w_lD

S 2

^¸

ab‚yDz

±

ubDet[q\e/ft[me[lGuSq_ª?molDne7Ì

(i, j)

^mol

H(I +hf i)

mz2t[q_lDz}u®w_lGuÆw_lDe[€‚yw_pz 5

a i,j −a i−α,j−β

^±_¬

bnmotb

txw.l²~?e%„e

¬

„}mu}u®e[l)w_z5

α − 1

X

p=0

(a _i − p,j − a _i − p − 1,j ) + X β−1 p=0

(a _i − α,j − p − a _i − α,j − p − 1 )

e[dÃw_„ ,lDq

¬%±

ubw.u…u®bne³D„z}u…ue[„d q_ª§ubDevw_~?q£š_ezyDde1€‚yw_poz

α.S 1

^¬

bDmope!u®bDez}e[t1qslDÀe1€‚yw_poz

β.S 2

^¸

e[dÃw_„ ,Íu®bw‡umª

(I + hX − 1i + hL − 1i) ∩ Q[X 1 , . . . , X n ]

^{« „e1zŽ ¸}

(I + hX − 1i + hL − 1i) ∩ Q [` 1 , . . . , ` k ]

^¯ mozÁ1e[„}q.·DmdÃe1lDzmqslw_p?mol

Q [X 1 , . . . , X n ]

^{« „}e[z}Ž ¸}

Q [` 1 , . . . , ` k ]

^¯7±

deg(I + hui + hX − 1i + hL − 1i) = 0

^{«„e[z}Ž ¸}

deg(I + hvi + hX − 1i + hL − 1i) = 0

^{¯ ¸}

Ø

ɟ

Ü 6º

I ⊂ R

^{7º( [½"} &3" &¹\º7»<º & $k½! \º (*)

D ⊂ N × N

^{º $ $¡º &¾ & )oº $}

(d, e)

$ ¾ %(

d ≤ n − 1

e ≤ k − 1

^(_»

d + e + 2

^{º -(*)$*º "(} s½#"(*) _½#"Àº7»%$7½'&‡»& º ( *"¶½$ $7½' )oº

?¼ ½#"(_¼ ¾ &3" &‡»<º7»#$ &

I

^{º (*)¹\º [¼4(_½¿¾ [½"} \º¿¹_¼®ºº&

I

^{\º7» & "º ~DmDe[r}

(I)

^{½ $ *º+$ "}

X

(d,e)∈D

deg(I + hu 1 i + · · · + hu d i + hv 1 i + · · · + hv e i + hX − 1i + hL − 1i)

º7¼®º

u 1 , . . . , u d , X

^{(˼®º $}

v 1 , . . . , v e , L

^* (_¼®º‰¹\º7»<º7¼ ½¿¾ )½­»<º (_¼ &3" &¹\º7»<º7» & $ A&‡¼?" $%½­» *º &3) "

» &3"¶½!(*){¼ ½­»\¹

Q[X 1 , . . . , X n ]

^{(˼®º $}

Q[` 1 , . . . , ` k ]

^*

º%zÅu®„q_lDr)w_porse1~D„®w.mot~DmηDe1rs„e1e &

I

^{½$*º $ "} &*º7(*)¹\º [¼4(_½¿¾ [½" \º¿¹_¼®ºº $&º ( *"¶½$ $7½' )oº

½$&3)(·º ?¼ ½#"(_¼¾ &3" &‡»<º7»#$ &

I

!²Ü Ÿ¡¢˜Ÿ Ü ž 5 l u®bDe´ze[€‚yDe1p

±

rsm¨šse[l wµ~Dm¨ÂËbDqsdfqsrse[lne[qsyDzfmonexw_p

I ⊂ R

^w_ln:w_l w_Ddfmoz}zm~Dpoe~DmÎÂ

DmdÃe1lDzmqsl

(d, e) ∈ N × N

^q_ª

I

^±n¬ ene[lDq_ue%~‚|

C d,e (I)

u®bDe%De1rs„e1eq_ª 5

I + hu 1 i + · · · + hu d i + hv 1 i + · · · + hv e i + hX − 1i + hL − 1i

¬

bDe1„e

u ₁ , . . . , u _d , X

^{« „e1zŽ ¸}

v ₁ , . . . , v _e , L

^¯ w.„e%rse[lne[„mt%pomolnexw_„bDq_dÃqsr_e[lDe1lDqsyDzªÉqs„}dÃzmolu®bneŽ?qsp¨|‚ lDqsdfm©w.p„}molDr

Q[X 1 , . . . , X n ]

^{« „e1zŽ ¸}

Q[` 1 , . . . , ` k ]

^{¯ ¸} ÏÆ|+t[q_l‚š_e[lGu®mqsl

±

m¨ª

d > n − 1

^qs„

e > k − 1

^±

C d,e (I)

^{mozl‚yDpp ¸}

lÔg˜— ±

ubDe!w_y˜u®bDqs„yDze1z

lGu®e1„ze1t1umoqsl¶abDe1qs„}|u®qMŽD„q¡š_eu®bw‡u2molu®bDeÆtxw_z}e!q_ª“wM~Dm¨ÂˎD„q‡†}e1t1umšse…š.w_„me1u·|

V ⊂ P ⁿ⁻¹ (C) × P ^k−1 (C)

q_ª2~DmηDmdÃe1lDzmqsl

(0, 0)

^± ubDet[w_„nmolw_pmu·|Àq_ª

V

^«É¬ bDmtb´t[q_molDt1moDe1zMml)ubDmoz

txw.ze

¬

mu®b)u®bDe%~DmηDe1rs„e1eq.ª&u®bnemDexw_pne1³lDmlDr

V

^¯ mz~§qsynlDDe[²~G|u®bDet[pow_zz}motxw.p~Dm¨ÂËbDqsdfqsrse[lne[qsyDz Ïj1Á[qsynuv~?qsyDlD

¸

l Η£g3

±

ubDew.ynu®bDq_„z‰r_mšsewÃz}modfmop©w.„z}uw‡u®e[dfe[lGu‰yDzmlDrÀDe7ªÉqs„dÃw.umoqslu®e1tbDlDmo€‚yDe1z

¸‰Ä

ee Ì\u®e[lnXu®bDmz

z}u®w.u®e1dÃe1lGuMuqfubDezÅu®„}qslDrÀ~nm¨Â·ne[rs„}e[e%q_ªVwf~Dm¨ÂˎD„q£†}e[t1umš_e%š.w_„}moe7u·|

V ⊂ P ^{n − 1} (C) × P ^{k − 1} (C)

De1³DlDe[

~G|Íw~DmηbDqsdfqsrse1lDe[q_yDz‰Ž?qsp|\lDq_dÃmow_p6z}|\z}ue[d

S ⊂ R

^¸ aBqu®bDmzKe[lD

±{¬

e¶ªÉqsppoq

¬

¨—¡”

¬

bDmtbÍrsmš_e[z%w

zmdÃmp©w_„Sz}u®w.u®e1dÃe1lGuMmlu®bDe%bDqsdfqsrse1lDe[q_yDztxw_z}e

¸

  Ü Ü žA­ŸÜ 6º

I ⊂ R

^{7º ( [½"} &3" &¹\º7»<º & $X½! \º (*)

D

^{*º "(} s½#"(*)2 _½#"Àº7»%$7½'&‡» &X½$

( *"¶½$ $7½' )oº?¼ ½#"(_¼ +¾ &3" &‡»<º7»$ (_»

f ∈ R

7º7(Àrse[lne[„mt [½" &3" &¹\º7»<º & $ &3)‡» &3"¶½!(*) & [½"

\º¿¹_¼®ºº

(α, β )

^{(˽ º}

f

^{½$» & ( ‡º7¼ & " _½¡½} $&‡¼%½­»

R/C

^{A&‡¼ (_» ( *"¶½$ $7½' )oº} ?¼7½#"(_¼ ²¾ &3" &‡»<º7»

C

&

I

%(¡½­»\¹7"( s½#"(*) _½#"Àº7»%$7½'&‡» * &‡»%$7½! \º7¼

D ⊂ N × N

^{*º $ $¡º $ “¾ *%(}

(d, e) ∈ D

^½2(_» 7&‡» )½

d + e + 2

^{º -(*)$}

D − 1

^(_»

0 ≤ d ≤ n − 1

^(_»

0 ≤ e ≤ k − 1

^{º7» º [½" \º¿¹_¼®ºº &}

I + hf i

^{º %(*)$}

α P

(d,e)∈D C d+1,e (I)

+ β P

(d,e)∈D C d,e+1 (I)

Ƽ &&

‹qslDz}moDe1„wÑ~Dm¨ÂËbDqsdfqsrse[lne[qsyDzfmonexw_p

I

^¬ bDqsz}e+w_Ddfmoz}zmo~npoeŽD„}modÃw_„}| t1qsdÎ?qslDe1lGu®zÀbDw¡š_e dÃw‡Ì˜modÃw_p˜nmodfe[lDz}moqsl

D

^± wkrse1lDe[„}mote[pe[dfe[lGu

f

^ªÉqs„

I

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