In Theorem 3.2 we show that two homogeneous polynomials $f$ and $g$ having
isomorphic Milnor algebras are right-equivalent.Comment: 6 page

A. Dimca

C.G. Gibson

Imran Ahmed

J.N. Mather

English

arXiv

summary:We recall first Mather's Lemma providing effective necessary and sufficient conditions for a connected submanifold to be contained in an orbit. We show that two homogeneous polynomials having isomorphic Milnor algebras are right-equivalent

Ahmed, Imran

Institute of Mathematics AS CR, v. v. i.

Czechoslovak Mathematical JournalImran AhmedHomogeneous polynomials with isomorphic Milnor algebrasCzechoslovak Mathematical Journal, Vol. 60 (2010), No. 1, 125–131Persistent URL: http://dml.cz/dmlcz/140556Terms of use:© Institute of Mathematics AS CR, 2010Institute of Mathematics of the Czech Academy of Sciences provides access to digitizeddocuments strictly for personal use. Each copy of any part of this document must contain theseTerms of use.This document has been digitized, optimized for electronic delivery andstamped with digital signature within the project DML-CZ: The Czech DigitalMathematics Library http://dml.czCzechoslovak Mathematical Journal, 60 (135) (2010), 125–131HOMOGENEOUS POLYNOMIALS WITHISOMORPHIC MILNOR ALGEBRASImran Ahmed, Lahore(Received August 13, 2008)Abstract. We recall first Mather’s Lemma providing effective necessary and sufficientconditions for a connected submanifold to be contained in an orbit. We show that twohomogeneous polynomials having isomorphic Milnor algebras are right-equivalent.Keywords: Milnor algebra, right-equivalence, homogeneous polynomialMSC 2010 : 14E05, 32S30, 14L30, 14J17, 16W221. IntroductionIn this note we recall first Mather’s Lemma 2.4 providing effective necessary andsufficient conditions for a connected submanifold to be contained in an orbit. InTheorem 3.2 we show that two homogeneous polynomials f and g having isomorphicMilnor algebras are right-equivalent.This is similar to the celebrated theorem by Mather and Yau [4], saying that theisolated hypersurface singularities are determined by their Tjurina algebras. Ourresult applies only to homogeneous polynomials, but it is no longer necessary toimpose the condition of having isolated singularities at the origin.2. Preliminary resultsWe recall here some basic facts on semialgebraic sets, which are also called con-structible sets, especially in the complex case. For a more complete introductionsee [2, Chapter 1].Definition 2.1. Let M be a smooth algebraic variety over K (K = R or C asusual).125(i) Complex Case. A subset A ⊂ M is called semialgebraic if A belongs to theBoolean subalgebra generated by the Zariski closed subsets ofM in the Booleanalgebra P (M) of all subsets of M .(ii) Real Case. A subset A ⊂ M is called semialgebraic if A belongs to Booleansubalgebra generated by the open sets Uf = {x ∈ U : f(x) > 0} where U ⊂ Mis an algebraic open subset in M and f : M → R is an algebraic function, inthe Boolean algebra P (M) of all subsets of M .By definition, it follows that the class of semialgebraic subsets ofM is closed underfinite unions, finite intersections and complements. If f : M → N is an algebraicmapping between the smooth algebraic varietiesM and N and if B ⊂ N is semialge-braic, then clearly f−1(B) is semialgebraic in M . Conversely, we have the followingbasic result.Theorem 2.2 (Tarski-Seidenberg-Chevalley). If A ⊂ M is semialgebraic, thenf(A) ⊂ N is also semialgebraic.Next consider the following useful result.Proposition 2.3. Let G be an algebraic group acting (algebraically) on a smoothalgebraic varietyM . Then the corresponding orbits are smooth semialgebraic subsetsin M .Let m : G×M → M be a smooth action. In order to decide whether two elementsx0, x1 ∈ M areG-transversal, we try to find a path (a homotopy) P = {xt : t ∈ [0, 1]}such that P is entirely contained in a G-orbit. It turns out this naive approach worksquite well and the next result gives effective necessary and sufficient conditions fora connected submanifold (in our case the path P ) to be contained in an orbit.Mather’s Lemma 2.4 ([3]). Let m : G × M → M be a smooth action andP ⊂ M a connected smooth submanifold. Then P is contained in a single G-orbit ifand only if the following conditions are fulfilled:(a) Tx(G.x) ⊃ TxP , for any x ∈ P .(b) dimTx(G.x) is constant for x ∈ P .3. Main TheoremFor isolated hypersurface singularities, the following result was obtained by Matherand Yau, see [4].126Theorem 3.1. Let f, g : (Cn, 0) → (C, 0) be two isolated hypersurface singulari-ties having isomorphic Tjurina algebras T (f) ≃ T (g). Then fK∼ g, whereK∼ denotesthe contact equivalence.For arbitrary (i.e. not necessarily with isolated singularities) homogeneous poly-nomials we establish now the following result.Theorem 3.2. Let f, g ∈ C[x1, . . . , xn]d = Hd(n, 1;C) = Hd be two homogeneouspolynomials of degree d such that Jf = Jg. Then fR∼ g, whereR∼ denotes the rightequivalence.P r o o f. To prove this claim we choose an appropriate submanifold ofHd(n, 1;C)containing f and g and then apply Mather’s lemma to get the result. Here f, g ∈Hd(n, 1;C) are such that Jf = Jg. Set ft = (1 − t)f + tg ∈ Hd(n, 1;C). Considerthe R-equivalence action on Hd(n, 1;C) under the group Gl(n,C). By eq. (4.16) [1,p. 35], we have(3.1) Tft(Gl(n,C).ft) = C〈xj∂ft∂xi; i, j = 1, . . . , n〉.Now, note that the R.H.S of eq. (3.1) satisfies the relationC〈xj∂ft∂xi; i, j = 1, . . . , n〉⊂ Jft ∩ Hd.But Jft ∩ Hd ⊂ Jf ∩ Hd since∂ft∂xi= (1 − t)∂f∂xi+ t∂g∂xi∈ (1 − t)Jf + tJg = Jf (because Jf = Jg).So, we have the inclusion of finite dimensional C-vector spaces(3.2) Tft(Gl(n,C).ft) = C〈xj∂ft∂xi; i, j = 1, . . . , n〉⊂ Jf ∩ Hdwith equality for t = 0 and t = 1.Let us show that we have equality for all t ∈ [0, 1] except finitely many values.Clearly the dimension of the space Jf ∩Hd is at most n2. Let us fix {e1, . . . , em}a basis of Jf ∩ Hd, where m 6 n2. Consider the n2 polynomialsαij(t) = xj∂ft∂xi= xj[(1 − t)∂f∂xi+ t∂g∂xi]127corresponding to the generators of the space (3.1). We can express each αij(t),i, j = 1, . . . , n in terms of the above mentioned fixed basis as(3.3) αij(t) = φ1ij(t)e1 + . . . + φmij (t)em, ∀ i, j = 1, . . . , nwhere each φkij(t) is linear in t. Consider the matrix of transformation correspondingto the eqs. (3.3)(φmij (t))n2×m =φ111(t) φ211(t) . . . φm11(t)....... . ....φ11n(t) φ21n(t) . . . φm1n(t)φ121(t) φ221(t) . . . φm21(t)....... . ....φ1nn(t) φ2nn(t) . . . φmnn(t)having rank at most m. Note that the equality〈xj∂ft/∂xi〉= Jf ∩ Hd holds forthose values of t in C for which the rank of the above matrix is precisely m. Aswe know that the rank of the matrix of transformation is at most m, there must ben2 − m proportional rows. So, we have the m × m submatrix whose determinantis a polynomial of degree m in t and by the fundamental theorem of algebra it hasat most m roots in C, for which rank of the matrix of transformation will be lessthan m. Therefore, the above-mentioned equality does not hold for finitely manyvalues, say t1, . . . , tp where 1 6 p 6 m.It follows that the dimension of the space (3.1) is constant for all t ∈ C exceptfinitely many values {t1, . . . , tp}.For an arbitrary smooth pathα : C −→ C \ {t1, . . . , tp}with α(0) = 0 and α(1) = 1, we have the connected smooth submanifoldP = {ft = (1 − α(t))f(x) + α(t)g(x) : t ∈ C}of Hd. By the above, it follows dimTft((Gl(n,C).ft)) is constant for ft ∈ P .Now, to apply Mather’s lemma, we need to show that the tangent space to thesubmanifold P is contained in that to the orbit Gl(n,C).ft for any ft ∈ P . Oneclearly hasTftP = {ḟt = −α̇(t)f(x) + α̇(t)g(x) ∀ t ∈ C}.Therefore, by Euler’s formula 7.6 [1, p. 101], we haveTftP ⊂ Tft(Gl(n,C).ft).128By Mather’s lemma the submanifold P is contained in a single orbit. This impliesthat fR∼ g, as required. Corollary 3.3. Let f, g ∈ Hd(n, 1;C). If M(f) ≃ M(g) (isomorphism of gradedC-algebras) then fR∼ g.P r o o f. We show firstly that an isomorphism of graded C-algebrasϕ : M(g) =C[x1, . . . , xn]Jg≃−→ M(f) =C[x1, . . . , xn]Jfis induced by a linear isomorphism u : Cn −→ Cn such that u∗(Jg) = Jf . Considerthe following commutative diagram.00Jgu∗//iJfjC[x1, . . . , xn]u∗//pC[x1, . . . , xn]qM(g)ϕ≃//M(f)0 0We note that the isomorphism ϕ is a degree preserving map and each of theJacobian ideals Jf and Jg is generated by the homogeneous polynomials of degreed− 1. The cases d = 1 and d = 2 are special, and we can treat them easily. Assumefrom now on that d > 3. Define the morphism u∗ : C[x1, . . . , xn] → C[x1, . . . , xn] by(3.4) u∗(xi) = Li(x1, . . . , xn) = ai1x1 + . . . + ainxn, ai1, . . . , ain ∈ Cwhich is well defined by commutativity of the diagram below.xiu∗//_pLi_qx̂iϕ≃// L̂i129Let us prove that u∗ is an isomorphism. Consider the following commutativediagram at the level of degree d = 1.C[x1, . . . , xn]1u∗1//pC[x1, . . . , xn]1q(M(g))1ϕ1≃// (M(f))1Since here the Jacobian ideals Jf and Jg are generated by polynomials ofdegree > 2, therefore we have(M(g))1 = (M(f))1 = C[x1, . . . , xn]1.This implies that ϕ1 and u∗1are coincident. As ϕ is a given graded C-algebra iso-morphism, it follows that u∗1is also an isomorphism. Hence u∗ is an isomorphism.Next we show that u∗(Jg) = Jf .For every G ∈ Jg, we have u∗(G) ∈ Jf by the commutative diagram below.Gu∗//_pF = u∗(G)_q0̂ϕ// F̂ = 0̂This implies that u∗(Jg) ⊂ Jf . As u∗ is an isomorphism, therefore it is invertibleand by repeating the above argument for its inverse, we have u∗(Jg) ⊃ Jf .Thus, u∗ is an isomorphism with u∗(Jg) = Jf .By eq. (3.4), the map u : Cn → Cn can be defined byu(z1, . . . , zn) = (L1(z1, . . . , zn), . . . , Ln(z1, . . . , zn))where Li(z1, . . . , zn) = ai1z1 + . . . + ainzn.Note that u is a linear isomorphism by Proposition 3.16 [1, p. 23].In this way, we have shown that the isomorphism ϕ is induced by a linear isomor-phism u : Cn → Cn such that u∗(Jg) = Jf .Here u∗(Jg) = 〈g1 ◦ u, . . . , gn ◦ u〉 = Jg◦u, where gj = ∂g/∂xj.Therefore, Jg◦u = Jf ⇒ g ◦ uR∼ f . Now, g ◦ uR∼ g implies that gR∼ f . Remark 3.4. The converse implication, namelyfR∼ g ⇒ M(f) ≃ M(g)always holds (even for analytic germs f , g defining IHS), see [1, p. 90].130References[1] A. Dimca: Topics on Real and Complex Singularities. Fried. Vieweg & Sohn, Braun-schweig/Wiesbaden, 1987.[2] C.G. Gibson, K. Wirthmüller, A.A. Du Plessis, E. J. N. Looijenga: Topological Sta-bility of Smooth Mappings. Lect. Notes Math. Vol. 552. Springer, Berlin, 1976.[3] J.N. Mather: Stability of C∞-mappings. IV: Classification of stable germs byR-algebras. Publ. Math., Inst. Hautes Étud. Sci. 37 (1969), 223–248.[4] J.N. Mather, S. S. T. Yau: Classification of isolated hypersurface singularities by theirmoduli algebras. Invent. Math. 69 (1982), 243–251.Author’s address: I . A hm e d, COMSATS Institute of Information Technology, M.A.Jinnah Campus, Defence Road, off Raiwind Road Lahore, Pakistan, e-mail: iahmedthegr8@gmail.com.131

Homogeneous polynomials with isomorphic Milnor algebras

John N. Mather

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Stability of C∞ mappings, IV: Classification of stable germs by R-algebrasmappings, IV: Classification of stable germs by R-algebras

Homogeneous polynomials with isomorphic milnor algebras

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Homogeneous Polynomials with Isomorphic Milnor Algebras

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