A Remark on Polynomial Mappings from to

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1 Aled Mahemacs h:// ISSN Ole: ISSN Pr: A Remar o Polyomal Mags from o ad a Alcao of he Sofware Male Research Th Bch Thuy Nguye UNESP Uversdade Esadual Paulsa Júlo de Mesqua Flho São José do Ro Preo Brasl How o ce hs aer: Nguye T.B.T. (6) A Remar o Polyomal Mags from o ad a Alca- o of he Sofware Male Research. Aled Mahemacs h://dx.do.org/.436/am Receved: July 6 6 Acceed: Seember 6 6 Publshed: Seember 9 6 Coyrgh 6 by auhor ad Scefc Research Publshg Ic. Ths wor s lcesed uder he Creave Commos Arbuo Ieraoal Lcese (CC BY 4.). h://creavecommos.org/lceses/by/4./ Oe Access Absrac I [] we cosruc sgular varees assocaed o a olyomal mag : where such ha f s a local submerso bu s o a fbrao he he -dmesoal homology ad erseco homology (wh oal erversy) of he varey are o rval. I [] he auhors rove ha f here exss a so-called very good rojeco wh resec o he regular value of a olyomal mag : he hs value s a aycal value of f ad oly f he Euler characersc of he fbers s o cosa. Ths aer rovdes relaos of he resuls obaed he arcles [] ad []. Moreover we rovde some examles o llusrae hese relaos usg he sofware Male o comlee he calculaos of he examles. We rovde some dscussos o hese relaos. Ths aer s a examle for graduae sudes o aly a sofware ha hey sudy he graduae rogram advaced researches. Keywords Sgulares Ierseco Homology Polyomal Mags Bfurcao. Prelmares.. Ierseco Homology We brefly recall he defo of erseco homology; for deals we refer o he fudameal wor of M. oresy ad R. MacPherso [3] (see also [4]). Defo.. Le X be a m-dmesoal varey. A srafcao of X s he daa of a fe flrao DOI:.436/am Seember 9 6

2 T. B. T. Nguye X = X X X X = m m such ha for every he se S = X \ X s eher a emy se or a mafold of dmeso. A coeced comoe of S s called a sraum of X. We deoe by cl he oe coe o he sace L he coe o he emy se beg a o. Observe ha f L s a srafed se he cl s srafed by he coes over he sraa of L ad a addoal -dmesoal sraum (he verex of he coe). Defo.. A srafcao of X s sad o be locally oologcally rval f for every x X \ X here s a oe eghborhood U x of x X a srafed se L ad a homeomorhsm such ha h mas he sraa of (roduc srafcao). h : U ; cl x U x (duced srafcao) oo he sraa of ( ;) cl The defo of erverses has orgally bee gve by oresy ad MacPherso: Defo.3. A erversy s a ( m + ) -ule of egers = ( 3 m ) such ha = = = ad { + + } for. Tradoally we deoe he zero erversy by = ( ) he maxmal erversy by = ( m ) ad he mddle erverses by m m m = (lower mddle) ad = (uer mddle). We say ha he erverses ad q are comlemeary f + q =. Le X be a varey such ha X adms a locally oologcally rval srafcao. We say ha a -dmesoal subse Y X s ( ) -allowable f dm Y X + for all. m Defe IC ( X ) o be he -vecor subsace of ξ such ha ξ s ( ) -allowable ad ξ s ( ) Defo.4. The C X cossg he chas -allowable. h erseco homology grou wh erversy deoed IH X wll refer o he erseco homology wh comac su- h by IH ( X ) s he homology grou of he cha comlex IC* ( X ). c The oao * cl ors ad he oao * suors. I he comac case hey cocde ad wll be deoed by IH* eral whe we wre H* ( X ) (res. IH* IH X wll refer o he erseco homology wh closed X. I ge- X ) we mea he homology (res. he erseco homology) wh boh comac suors ad closed suors. oresy ad MacPherso roved ha he erseco homology s deede o he choce of he srafcao sasfyg he locally oologcally rval codos. The Pocaré dualy holds for he erseco homology of a (sgular) varey: Theorem.5. (oresy MacPherso [3]) For ay oreable comac srafed sem-algebrac m-dmesoal varey X he geeralzed Pocaré dualy holds: q IH X IH X m where ad q are comlemeary erverses. For he o-comac case we have: 869

3 T. B. T. Nguye q cl IH X IH X c m.. The Bfurcao Se he Se of Asymoc Crcal Values ad he Asymoc Se m Le : where m be a olyomal mag. ) The bfurcao se of deoed by B( ) s he smalles se s o C -fbrao o hs se (see for examle [5]).. m such ha ) Whe = m we deoe by S he se of os a whch he mag s o roer.e. m { α { } α} S : = : z z such ha z B ad call he asymoc varey (see [6]). The followg holds:. Varees Assocaed o a Polyomal Mag : I [] we cosruc sgular varees assocaed o a olyomal mag as follows: le : s he se of crcal values of. Le ρ : where a = such ha K = where K be a real fuco such ha ρ = a z + + a z a ad a. Le us deoe ϕ as a real mag from o. Le us defe ρ ( ϕ) ( ϕ) = S ([6]). : = ad cosder ( ϕ ) + { } : = Sg = x such ha Ra D x where D( ϕ )( x) s he (real) Jacoba marx of ( ϕ ) : ha Sg ( ϕ) = Sg ( ρ) so we have = Sg ( ρ ). : a x. Noce Prooso.. [] For a oe ad dese se of olyomal mags such ha K = he varey s a smooh mafold of dmeso. Now le us cosder: a) F : = he resrco of o b) \ F = ( K ( F )). Sce he dmeso of s (Prooso.) he locally a egh- bourhood of ay o x we ge a mag F :. The here exss a coverg { U U} of by oe sem-algebrac subses ( ) such ha o every eleme of hs coverg he mag F duces a dffeomorhsm oo s mage (see Lemma. of [7]). We ca fd sem-algebrac closed subses V U ( ) whch cover as well. Thas o Mosows s Searao Lemma (see Searao Lemma [7]. 46) for each = here exss a Nash fuco ψ : such ha ψ s osve o V ad egave o \ U. We ca choose he Nash fucos x x ψ such ha ψ eds o zero whe { } 87

4 T. B. T. Nguye eds o fy. Le he Nash fucos zero ad ( x ) assocaed o ( ) ψ ad ρ be such ha ( x ) ψ eds o ρ eds o fy whe x eds o fy. Defe a varey ρ as ( F ψ ψ )( ) : = ha meas s he closure of F ψ ψ. I order o udersad beer he cosruco of he varey 4.3 []. Prooso.. [] Le K = ad le ρ : : by see he examle be a olyomal mag such ha be a real fuco such ha ρ = a z + + a z where a = a ad a for =. The here exss a real algebrac varey where > such + ha: ) The real dmeso of s ) The sgular se a fy of he varey where s coaed ( ρ ) { } { } = { z } Sg ( ) z ( z ) ρ : α ρ : eds o fy eds o α. 3. The Bfurcao Se B ad he Homology Ierseco Homology of Varees Mag : Assocaed o a Polyomal We have he wo followg heorems dealg wh he homology ad erseco homology of he varey. Theorem 3.. [] Le ( ) 3 K ( ) =. If B he ) H ( ) ) IH ( ) Theorem 3.. [] Le = ( ) : = : be a olyomal mag such ha where s he oal erversy. where 4 be a olyomal mag such ha K ( ) = ad Ra ( ˆ D ) where ˆ s he = leadg form of ha s he homogeous ar of hghes degree of for = B he. If ) H ( ) ) H 4 ( ) 3) IH where s he oal erversy. Remar 3.3. The sgular se a fy of deeds o he choce of he fuc- also chages. However we have alway o ρ sce whe ρ chages he se he roery B ( ρ ) (see [8]). Remar 3.4. The varey deeds o he choce of he fuco ρ ad he fucos ψ bu he heorems 3. ad 3. do o deed o he varees. Form ow we deoe by ( ρ ) ay varey assocaed o ( ρ ). If we refer o 87

5 T. B. T. Nguye ha meas a varey assocaed o ( ) ρ for ay ρ. 4. The Bfurcao Se B ad he Euler Characersc of he Fbers of a Polyomal Mag Le : = : be a o-cosa olyomal mag ad ( = ) be a regular value of. Defo 4.. [] A lear fuco L : s sad o be a very good rojeco wh resec o he value ( ) { : δ} D = = < : δ ) The resrco ) The cardal of Theorem 4.. [] Le f here exss a osve umber δ such ha for all L : = L s roer L λ does o deed o λ where λ s a regular value of L. good rojeco wh resec o he value oly f he Euler characersc of ( ) be a regular value of. Assume ha here exss a very. The Theorem 4.3. [] Assume ha he zero se s a aycal value of f ad s bgger ha ha of he geerc fber. z : ˆ z = = where { } ˆ s he leadg form of mag L s a very good rojeco wh resec o ay regular value 5. Relaos bewee [] ad [] Le = ( ) : ( 3) K ( ) =. The ay has comlex dmeso oe. The ay geerc lear of. be a olyomal mag such ha s a regular value of. Le ρ : be a real fuco such ha ρ = a z + + a z where a = a ad a for =. From heorems 3. ad 4. we have he followg corollary. K = : be a olyomal mag such ha 3 Corollary 5.. Le ( ) =. Assume ha here exss a very good rojeco wh resec o he Euler characersc of ( ) ) H ( ( ρ ) ) ) IH ( ( ρ ) ) for ay ρ s bgger ha ha of he geerc fber he for ay ρ where s he oal erversy.. If 3 Proof. Le = ( ) : be a olyomal mag such ha The every o rojeco wh resec o K =. s a regular o of. Assume ha here exss a very good. If he Euler characersc of ( ) s bgger ha ha of he geerc fber he by he heorem 4. he bfurcao se B( ) s o emy. The by he heorem 3. we have H ( ( ρ ) ) for ay ρ ad ( ) IH ρ for ay ρ where s he oal erversy. From heorems 3. ad 4. we have he followg corollary. Corollary 5.. Le = ( ) : where 4 be a olyomal mag such ha K ( ) = ad Ra ( ˆ D ) where ˆ s he = leadg form of. Assume ha here exss a very good rojeco wh resec o. If he Euler characersc of ( ) he ) ( ) H ρ for ay ρ s bgger ha ha of he geerc fber 87

6 T. B. T. Nguye ) H4 ( ( ρ ) ) for ay ρ 3) IH ( ) Proof. Le = ( ) : such ha for ay ρ where s he oal erversy. where 4 be a olyomal mag K =. The every o s a regular o of. Assume ha here exss a very good rojeco wh resec o. By he heorem 4. Ra Dˆ he by he he bfurcao se heorem 3. we have ) H ( ( ρ ) ) ) H ( ( ρ ) ) 3) IH B s o emy. If ( ) = for ay ρ 4 for ay ρ for ay ρ where s he oal erversy. We have also he followg corollary. = : where 4 be a olyomal ma- z : z = = Corollary 5.3. Le ( ) g such ha K ( ) =. Assume ha he zero se ˆ { } has comlex dmeso oe where ˆ s he leadg form of. If he Euler charac- s bgger ha ha of he geerc fber where he ersc of ) H ( ( ρ ) ) ) H ( ( ρ ) ) 3) IH ( ( ρ ) ) Proof. A frs sce he zero se { } for ay ρ 4 for ay ρ for ay ρ where s he oal erversy. z : ˆ z = = has comlex dmeso oe he by he heorem 4.3 ay geerc lear mag L s a very good rojeco wh resec o ay regular value of. Moreover he comlex dmeso of he se { z : ˆ z = = } s he comlex cora of D Ra Dˆ =. By he corollary 5. we ge he roof ( ˆ ). The = ( ) of he corollary 5.3. = Remar 5.4. We ca cosruc he varey ( L ) defed 4. as he followg: Le = ( ) : olyomal mag such ha K =. Assume ha here exss a very good rojeco L : L wh resec o = az =. The he varey ( L ) wh a hs varey varees ( L ) where L s a very good rojeco where be a. The L s a lear fuco. Assume ha s defed as he varey ( ρ ) ρ = a z = where z are he modules of he comlex umbers a ad z resecvely. Wh L all he resuls he corollares ad 5.3 hold. Moreover he mae he corollares ad 5.3 smler. Remar 5.5. I he cosruco of he varey by he resrco of ( ϕ ) o ha meas ( ϕ ) F : = [] (see seco ) f we relace F he we have he same resuls ha []. I fac hs case sce he dmeso of s he locally a eghbourhood of ay o x mag F : U U sem-algebrac subses (. There exss also a coverg { } we ge a by oe of ) such ha o every eleme of hs coverg he ma- 873

7 T. B. T. Nguye g F duces a dffeomorhsm oo s mage. We ca fd sem-algebrac closed subses V U ( \ U ) whch cover as well. Thas o Mosows s Searao Lemma for each = here exss a Nash fuco ψ : such ha ψ s osve o V ad egave o \ U. Le he Nash fucos ψ ad ρ be such ha ψ ( z) ad ϕ ( z ) = + ρ z ed o zero where { z } s a sequece edg o fy. Defe a varey assocaed o ( ) ( F ψ ψ ) ( ϕ ψ ψ ) : = =. We ge he ( ) -dmesoal sgular varey a fy of whch s { + } Wh hs cosruco of he se 6. Some Dscussos. + ρ as he sgular se he corollares ad 5.3 also hold. A aural queso s o ow f he coverses of he corollares 5. ad 5. hold. Tha meas le = ( ) : ( 3) be a olyomal mag such ha K ( ) = he Queso 6.. If here exss a very good rojeco wh resec o ad f eher IH ( ) or H ( ) he s he Euler characersc of ( ) bgger ha he oe of he geerc fber? By he heorem 4. he above queso s equvale o he followg queso: Queso 6.. If B( ) = he are IH ( ) = ad IH ( ) =? Ths queso s equvale o he coverse of he heorems 3. ad 3.. Noe ha by he rooso. he sgular se a fy of he varey s coaed ( ρ ) { + }. Moreover he roofs of he heorems 3. ad 3. we see ha he characerscs of he homology ad erseco homology of he varey ( ρ ) deed o he se ( ρ ). I [] we rovded a examle o show ha he aswer o he queso 6. s egave. I fac le he K ( ) = ad. ( ρ ) = ad ( ) ( ρ ) = ad ( ) ( ζ ) ( ζ ) zw = zz + w 3 : B = f we choose he fuco ρ = ζ he IH ρ = ; f we choose he fuco ρ = w he IH ρ. The we sugges he wo followg cojecures. Cojecure 6.3. Does here exs a fuco ρ such ha f B( ) = he ( ρ ) =? Cojecure 6.4. Le = ( ) : g such ha K sec o. If he Euler characersc of s cosa for ay he here exss a real osve fuco ρ : such ha ( ) ad IH ( ( ρ ) ) =. be a olyomal ma- =. Assume ha here exss a very good rojeco wh re- H ρ = Remar 6.5. The cosruco of he varey [] (see seco ) ca be aled 874

8 T. B. T. Nguye m for ay olyomal mag : where m such ha K =. I fac f s geerc he smlarly o he rooso. he varey { } ( ϕ) ( ϕ) : = Sg = x such ha Ra D x m has he real dmeso m. Hece f we cosder F : ha meas F s he resrco of o = he locally we ge a real mag B ρ hs case we also have F : for ay ρ (see [8]) where m = { z } Sg ( ) z ( z ) m m. Moreover { } ρ : α ρ : eds o fy eds o α. So we ca use he same argumes [] ad we have he followg resuls. m Prooso 6.6. Le : be a olyomal mag where m such ha K =. Le ρ : be a real fuco such ha ρ = a z + + a z where a = a ad a for =. The here exss a real varey where > such m+ ha: ) The real dmeso of s m ) The sgular se a fy of he varey s coaed ( ρ ) { }. Smlarly o [] we have he wo followg heorems (see heorems 3. ad 3.). = : where 4 be a olyomal mag Theorem 6.7. Le ( ) such ha K ( ) =. If B he ) H ( ) ) IH ( ) Theorem 6.8. Le = ( m ) mag such ha he leadg form of. If B he ) H ( ( ρ ) ) for ay ρ ) H4 ( ( ρ ) ) for ay ρ 3) IH ( ( ρ ) ) 7. Examles where s he oal erversy. : m where 3 m be a olyomal Ra Dˆ m where ˆ s K =. Assume ha ( ) = for ay ρ where s he oal erversy. Examle 7.. We gve here a examle o llusrae he calculaos of he se he case of a olyomal mag : where K ( ) = here exss a very good rojeco wh resec o ay o of m B ad B. I geeral he calculaos of he se are eough comlcae bu he sofware Male may suor us. Tha s wha we do hs examle. Le us cosder he Brougho s examle [9]: zw z zw : = +. We have K ( ) = ad B. I fac sce he sysem of equaos δ δ = = has o soluos he K ( ) =. Moreover δz δw 875

9 T. B. T. Nguye ad for ay we have = { = = } ( { }) z w : z or zw \ { } { } ( ) = zw : z ad w= z z \. So s o homeomorhc o for ay. Hece { } B =. We deerme ow all he ossble very good rojecos of wh resec o = B. I fac for ay δ > ad for ay D δ we have z = { : + = } = ( ) : ad =. z zw z zw zw z w Assume ha {( z w ) } s a sequece fy he w eds o zero. If very good rojeco wh resec o w eds o fy he edg o fy. If z eds o z eds o zero. If L s a = he by defo he resrco L : = L s roer. The L = az + bw where a ad b. We chec # L λ of L ( λ ) where λ s a regular value of L. Le us relace z w = he equao az + bw = λ we have he followg equao z ow he cardal z az + b = z where z. Ths equao always has hree (comlex) soluos. Thus he umber # L λ does o deed o λ. Hece ay lear fuco of he form L = az + bw where a ad b s a very good rojeco of wh resec o =. I s easy o see ha he se of very good rojecos of wh resec o he se of lear fucos. We choose L = z+ w ad we comue he varey ρ = z + w. Le us deoe z = x + x w = x + x 3 4 = s dese assocaed o ( ) where x x x3 x4. Cosder as a real olyomal mag we have ad ( 3 4) = ( ) x x x x x xx xx xxx x xxx xx xx The se = Sg ( ρ ) ϕ = = =. + ρ z x + x + x + x + w 3 4 ρ where s he se of he soluos of he deerma of he mors 3 3 of he marx + xx 3xx 4 xx 3xx 4 xx xx D ( ρ ) = xx 3+ xx 4 + xx 3xx 4 xx xx. x x x3 x 4 Usg Male we: A) Calculae he deermas of he mors 3 3 D ρ : ) Calculae he deerma of he mor defed by he colums ad 3: of he marx 876

10 T. B. T. Nguye ) Calculae he deerma of he mor defed by he colums ad 4: 3) Calculae he deerma of he mor defed by he colums 3 ad 4: 877

11 T. B. T. Nguye 4) Calculae he deerma of he mor defed by he colums 3 ad 4: B) Solve ow he sysem of equaos of he above deermas: We coclude ha = N N N3 where 3 N = x = x x = x = RooOf _ Z x + x _ Z x = { } { (_ ) } { x4 = RooOf ( x + x) _ Z x x + _ Z x 3 4 N = x = x x = RooOf Z + x x = x = 3 4 } N = x = x x = x x = RooOf x + x _ Z x x + _ Z x 3 3 C) I order o calculae = 3. ) Calculae ad draw ( )( N ) we have o calculae ad draw ( ϕ )( N ) ϕ : for 878

12 T. B. T. Nguye ) Calculae ad draw ( )( N ) ϕ : + Calculae ad draw ( )( N ) ϕ : 3 879

13 T. B. T. Nguye Sce 3 s he closure of ( ϕ ) N = he s coeced ad has a ure dmeso he V F s a coe: 88

14 T. B. T. Nguye Examle 7.. If we ae he suseso of he Brougho s examle ( η) ( η) zw = z+ zw 3 : he smlarly o he examle 7. he varey s a coe as he examle 7. bu 6 has dmeso 4 he sace. We ca chec easly ha he erseco homology dmeso of he varey llusrae he corollary 5.. of hs examle s o-rval. We ge a examle o 3 Examle 7.3. If we ae he Brougho examle for : such ha ( zw η ) = z+ zw he smlarly o he examle 7. we ge a examle of varees for he case : m where m. Ths examle llusraes he remar 6.5. Refereces [] Nguye T.B.T. ad Ruas M.A.S. (5) O a Sgular Varey Assocaed o a Polyomal Mag. ArXv: [] Ha V.H. ad Nguye T.T. () O he Toology of Polyomal Mags from. Ieraoal Joural of Mahemacs h://dx.do.org/.4/s967x684 o [3] oresy M. ad MacPherso R. (98) Ierseco Homology Theory. Toology h://dx.do.org/.6/4-9383(8)93-8 [4] Brassele J.-P. (5) Iroduco o Ierseco Homology ad Perverse Sheaves. Isuo de Maemáca Pura e Alcada Ro de Jaero. [5] Kurdya K. Orro P. ad Smo S. () Semalgebrac Sard Theorem for eeralzed Crcal Values. Joural of Dffereal eomery [6] Jeloe Z. (993) The Se of Po a Whch Polyomal Ma Is No Proer. Aales Poloc Mahemac [7] Mosows T. (976) Some Proeres of he Rg of Nash Fucos. Aal della Scuola ormale suerore d Psa [8] Das L.R.. Ruas M.A.S. ad Tbar M. () Regulary a Ify of Real Mags ad a Morse-Sard Theorem. Joural of Toology h://dx.do.org/./jool/js5 [9] Brougho A. (98) O he Toology of Polyomal Hyersurfaces. Proc. Symos. Pure Mah. Vol

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