Direct sum decomposability of polynomials and factorization of associated forms
aa r X i v : . [ m a t h . AG ] N ov DIRECT SUM DECOMPOSABILITY OF POLYNOMIALS ANDFACTORIZATION OF ASSOCIATED FORMS
MAKSYM FEDORCHUK
Abstract.
We prove two criteria for direct sum decomposability of homogeneouspolynomials. For a homogeneous polynomial with a non-zero discriminant, weinterpret direct sum decomposability of the polynomial in terms of factorizationproperties of the Macaulay inverse system of its Milnor algebra. This leads to anif-and-only-if criterion for direct sum decomposability of such a polynomial, andto an algorithm for computing direct sum decompositions over any field, either ofcharacteristic 0 or of sufficiently large positive characteristic, for which polynomialfactorization algorithms exist.For homogeneous forms over algebraically closed fields, we interpret direct sumsand their limits as forms that cannot be reconstructed from their Jacobian ideal.We also give simple necessary criteria for direct sum decomposability of arbitraryhomogeneous polynomials over arbitrary fields and apply them to prove that manyinteresting classes of homogeneous polynomials are not direct sums. Introduction
A homogeneous polynomial f is called a direct sum if, after a change of variables,it can be written as a sum of two or more polynomials in disjoint sets of variables:(1.1) f = f ( x , . . . , x a ) + f ( x a +1 , . . . , x n ) . Homogeneous direct sums are the subject of a well-known symmetric Strassen’s ad-ditivity conjecture postulating that the Waring rank of f in (1.1) is the sum of theWaring ranks of f and f (see, for example, [15]). Direct sums also play a specialrole in the study of GIT stability of associated forms [9].The innocuous definition of a direct sum raises several natural questions: How dowe determine whether a given polynomial is a direct sum? For example, is f = x + 3 x x + 3 x x + 2 x + 3 x x + 6 x x x + 4 x x + 3 x x + 4 x x + 2 x a direct sum in Q [ x , x , x ]? (See Example 6.4 for the answer). Does the locusof direct sums in the space of all homogeneous polynomials of a given degree has ageometric meaning?In this paper, we answer these questions by considering two natural maps on thespace of forms of a given degree, the gradient morphism ∇ and the associated formmorphism A , described in more detail later on. Our first main result is a new criterion Hereinafter, we refer to any homogeneous polynomial as a form, and we call a form f in n variables smooth if it defines a smooth hypersurface in P n − ; see § for recognizing when a smooth form is a direct sum over a field either of characteristic0 or of sufficiently large positive characteristic: Theorem A (see Theorem 4.1) . A smooth form f is a direct sum if and only if itsassociated form A ( f ) is a nontrivial product of two factors in disjoint variables. Our second main result is the characterization of the locus of direct sums as thenon-injectivity locus of ∇ , which, as will be clear, is the locus where ∇ has positivefiber dimension. While the full statement of this result given by Theorem 3.2 is toocumbersome to state in the introduction, it is well illustrated by the following: Theorem B (see Theorem 3.2) . Suppose f is a GIT semistable form over alge-braically closed field of characteristic not dividing deg( f )! . Then f is a direct sum ifand only if there exists g , which is not a scalar multiple of f , and such that ∇ f = ∇ g . The problem of finding a direct sum decomposability criterion for an arbitraryhomogeneous polynomials has been successfully addressed earlier by Kleppe [12] overan arbitrary field, and Buczy´nska-Buczy´nski-Kleppe-Teitler [4] over an algebraicallyclosed field. Both works interpret direct sum decomposability of a form f in termsof its apolar ideal f ⊥ (see § f ⊥ to define an associative algebra M ( f ) of finite dimension over thebase field. He then proves that, over an arbitrary field, direct sum decompositions of f are in bijection with complete sets of orthogonal idempotents of M ( f ). Buczy´nska,Buczy´nski, Kleppe, and Teitler prove that for a form f of degree d over an algebraicallyclosed field, the apolar ideal f ⊥ has a minimal generator in degree d if and only ifeither f is a direct sum, or f is a highly singular polynomial. In particular, over analgebraically closed field, [4] gives an effective criterion for recognizing when f is adirect sum in terms of the graded Betti numbers of f ⊥ . We will show that TheoremB is essentially equivalent to the main result of [4], thus giving a different proof ofthis result.However, none of the above-mentioned two works seem to give an effective methodfor computing a direct sum decomposition when it exists, and the criterion of [4] can-not be used over non-closed fields (see Example 6.5). A key step in the proof of thedirect sum criterion in [4] is the Jordan normal form decomposition of a certain linearoperator, which in general requires solving a characteristic equation. Similarly, findinga complete set of orthogonal idempotents requires solving a system of quadratic equa-tions. This makes it challenging to turn [4] or [12] into an algorithm for finding directsum decompositions when they exist. Although our criterion given by Theorem Aworks only for smooth forms, it does so over an arbitrary field either of characteristic0 or of sufficiently large characteristic, and it leads to an algorithm for finding directsum decompositions over any such field for which polynomial factorization algorithmsexist. This algorithm is given in Section 6.Recall that to a smooth form f of degree d + 1 in n variables, one can assigna degree n ( d −
1) form A ( f ) in n (dual) variables, called the associated form of f ([1, 2, 5]). The associated form A ( f ) is defined as a Macaulay inverse system of theMilnor algebra of f [1], which simply means that the apolar ideal of A ( f ) coincides ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 3 with the Jacobian ideal of f : A ( f ) ⊥ = ( ∂f /∂x , . . . , ∂f /∂x n ) . Such definition leads to an observation that for a smooth form f that is written as a sum of two forms in disjoint sets of variables, the associated form A ( f ) decomposesas a product of two forms in disjoint sets of (dual) variables ([9, Lemma 2.11]). Forexample, up to a scalar, A ( x d +11 + · · · + x d +1 n ) = z d − · · · z d − n . The main purpose of Theorem A is to prove the converse statement, and thus establishan if-and-only-if criterion for direct sum decomposability of a smooth form f in termsof the factorization properties of its associated form A ( f ) (see Theorem 4.1).In Lemma 2.3, we give a simple necessary condition, valid over an arbitrary field,for direct sum decomposability of an arbitrary form in terms of its gradient point . Itis then applied in Section 5 to prove that a wide class of homogeneous forms containsno direct sums.1.1. Notation and conventions.
Throughout, k will be a field. Unless stated oth-erwise, we do not require k to be algebraically closed or to be of characteristic 0.If U is a k -vector space, and W is a subset of U , we denote by h W i the k -linearspan of W in U .If W is a representation of the multiplicative group scheme G m = Spec k [ t, t − ],then, for every i ∈ Z , we denote by W ( i ) the weight-space in W of the G m -action ofweight i . We then have a decomposition W = M i ∈ Z W ( i ) . We fix an integer n ≥ k -vector space V of dimension n . We set S :=Sym V , and D := Sym V ∨ to be the symmetric algebras on V and V ∨ , respectively,with the standard grading. If x , . . . , x n is a basis of V , then we identify S withthe usual polynomial ring k [ x , . . . , x n ]. If z , . . . , z n is the dual basis of V ∨ , then D = k [ z , . . . , z n ] is the graded dual of S . Homogeneous elements of S and D willbe called forms . If f = 0 ∈ S d is a degree d form, then we will denote by ¯ f thecorresponding element of the projective space P S d .For a homogeneous ideal I ⊂ S , we denote by V ( I ) the closed subscheme of P V ∨ ≃ P n − defined by I . In particular, a nonzero form f ∈ S d defines a hypersurface V ( f )in P V ∨ . If k is algebraically closed, the hypersurface V ( f ) determines ¯ f , and so wewill sometimes not distinguish between a hypersurface and its equation. With thisin mind, we say that a form f ∈ S d +1 is smooth if the hypersurface V ( f ) ⊂ P n − is smooth over k (this is, of course, equivalent to V ( f ) being non-singular over thealgebraic closure of k ). The locus of smooth forms in P S d +1 will be denoted by( P S d +1 ) ∆ .We have a differentiation action of S on D , also known as the polar pairing . Namely,if x , . . . , x n is a basis of V , and z , . . . , z n is the dual basis of V ∨ , then the pairing MAKSYM FEDORCHUK S × D → D is given by g ◦ F := g ( ∂/∂z , . . . , ∂/∂z n ) F ( z , . . . , z n ) , for g ∈ S and F ∈ D .Given a non-zero form F ∈ D d , the apolar ideal of F is defined to be F ⊥ := { g ∈ S | g ◦ F = 0 } ⊂ S. We let the space of essential variables of F to be E ( F ) := { g ◦ F | g ∈ S d − } ⊂ D . If char( k ) ∤ d !, then the pairing S d × D d → k is perfect, and for every F ∈ D d , wehave F ∈ Sym d E ( F ) ⊂ D d (see [4, § F can be expressed as apolynomial in its essential variables. This property explains the name of E ( F ).Importantly, working under the assumption that char( k ) ∤ d !, the graded k -algebra S/F ⊥ is a Gorenstein Artin local ring with socle in degree d . In fact, a well-knowntheorem of Macaulay establishes a bijection between graded Gorenstein Artin quo-tients S/I of socle degree d and elements of P D d (see, e.g., [11, Lemma 2.12] or [7,Exercise 21.7]). Definition 1.2 (Gradient morphism) . Let x , . . . , x n be a basis of V . For f ∈ S d +1 ,we define h∇ f i := h ∂f /∂x , . . . , ∂f /∂x n i ⊂ S d to be the span of all first-order partials of f . If dim k h∇ f i = n , we denote by ∇ f thepoint of Grass( n, S d ) corresponding to h∇ f i , and call ∇ f the gradient point of f .The Jacobian ideal of f is defined to be J f := ( ∇ f ) = ( ∂f /∂x , . . . , ∂f /∂x n ) ⊂ S, and the Milnor algebra of f is M f := S/J f .Following the terminology of [4, § f ∈ S d (respectively, ¯ f ∈ P S d ) concise if it cannot be written as a form in less than n = dim k V variables, or equivalently if f ∈ Sym d W for W ⊂ V implies that W = V . If char( k ) ∤ d !, then f ∈ S d is conciseif and only if E ( F ) = V if and only if dim k h∇ f i = n . We set P ( S d ) c := { ¯ f ∈ P S d | dim k h∇ f i = n } . Clearly, P ( S d ) c is an open subset of P S d . If char( k ) ∤ d !, then P ( S d ) c is the locus ofconcise forms, and so ¯ f ∈ P S d \ P ( S d ) c if and only if the hypersurface defined by f isa cone if and only if E ( f ) ( V . Remark 1.3.
Even though we allow k to have positive characteristic, we do nottake D to be the divided power algebra (cf. [11, Appendix A]), as the reader mighthave anticipated. The reason for this is that at several places we cannot avoid but toimpose a condition that char( k ) is large enough (or zero). In this case, the dividedpower algebra is isomorphic to D up to the degree in which we work. But note that f = x + yz is always concise in k [ x, y, z ] while dim k h∇ f i = 2 < k ) = 2. ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 5
Direct sums and products.
In this subsection, we recall a well-known defini-tion of a direct sum polynomial, and introduce more exotic notions of a direct productpolynomial, and a direct sum space of polynomials.1.2.1.
Direct sum forms.
A form f ∈ Sym d +1 V is called a direct sum if there is adirect sum decomposition V = U ⊕ W and nonzero f ∈ Sym d +1 U and f ∈ Sym d +1 W such that f = f + f . In other words, f is a direct sum if and only if for some choiceof a basis x , . . . , x n of V , we have that f = f ( x , . . . , x a ) + f ( x a +1 , . . . , x n ) , where 1 ≤ a ≤ n −
1, and f , f = 0. Remark 1.4.
Note that the roles of S and D are interchangeable in § f ∈ S , we can define the apolar ideal f ⊥ ⊂ D and the space of essential variables E ( f ) ⊂ S . With this notation, if char( k ) ∤ ( d + 1)!, then f ∈ S d +1 is a direct sumif and only if we can write f = f + f , where f , f = 0 and E ( f ) ∩ E ( f ) = 0.Furthermore, under the same assumption on char( k ), we have that dim k h∇ f i =dim k E ( f ).1.2.2. LDS forms.
We say that f ∈ S d +1 is an LDS form (or, simply, f is LDS) if,after a linear change of variables, we can write(1.5) f ( x , . . . , x n ) = ℓ X i =1 x i ∂H ( x ℓ +1 , . . . , x ℓ ) ∂x ℓ + i + G ( x ℓ +1 , . . . , x n ) , where H and G are degree d + 1 forms, in ℓ and n − ℓ variables, respectively. Toour knowledge, LDS forms first appeared in [4, § f from (1.5) satisfies(1.6) f = lim t → t (cid:2) H ( tx + x ℓ +1 , . . . , tx ℓ + x ℓ ) − H ( x ℓ +1 , . . . , x ℓ )+ tG ( tx + x ℓ +1 , . . . , tx ℓ + x ℓ , x ℓ +1 , . . . , x n ) (cid:3) . Since the form on the right-hand side of the above equation is a direct sum for t = 0,we obtain an explicit presentation of the LDS form f as a limit of direct sums.Conversely, [4, Theorem 4.5] proves that every concise limit of direct sums that is notitself a direct sum is an LDS form. Lemma 1.7. If f is an LDS form of degree deg( f ) = d + 1 ≥ , then f is GITnon-semistable with respect to the standard SL( n ) -action on S d +1 .Proof. Suppose f is as in (1.5). Then with respect to the one-parameter subgroup ofSL( n ) acting on a basis x , . . . , x ℓ , x ℓ +1 , . . . , x ℓ , x ℓ +1 , . . . , x n of V with weights(1 + ǫ, , . . . , | {z } ℓ − , − , . . . , − | {z } ℓ , − ǫ/ ( n − ℓ ) , . . . , − ǫ/ ( n − ℓ )) , where 0 < ǫ ≪ , the maximum weight of a monomial in f ismax { (1 + ǫ ) − d, − ( d + 1) ǫ/ ( n − ℓ ) } < . Hence f is non-semistable by the Hilbert-Mumford numerical criterion. (cid:3) MAKSYM FEDORCHUK
Direct product forms.
By analogy with direct sums, we will call a nonzero form F ∈ D a direct product if there is a non-trivial direct sum decomposition V ∨ = U ⊕ W such that F = F F for some F ∈ Sym U and F ∈ Sym W . In other words, a nonzerohomogeneous F ∈ Sym D is a direct product if and only if for some choice of a basis z , . . . , z n of V ∨ , we have that(1.8) F ( z , . . . , z n ) = F ( z , . . . , z a ) F ( z a +1 , . . . , z n ) , where 1 ≤ a ≤ n − balanced if( n − a ) deg( F ) = a deg( F ) . Note that if char( k ) ∤ (deg F )!, then a non-trivial factorization F = F F is a directproduct decomposition if and only if E ( F ) ∩ E ( F ) = 0. This observation reducesthe problem of recognition of direct products to a factorization problem.1.2.4. Direct sum and decomposable spaces of forms.
Suppose L ⊂ Sym d V is a sub-space of dimension m . We say that [ L ] ∈ Grass( m, Sym d V ) is a direct sum ifthere is a non-trivial direct sum decomposition V = U ⊕ W and elements [ L ] ∈ Grass( m , Sym d U ) and [ L ] ∈ Grass( m , Sym d W ), where m + m = m , such that(1.9) L = L + L ⊂ Sym d U ⊕ Sym d W ⊂ Sym d V. We will further say that [ L ] is a balanced direct sum if m = dim k U and m = dim k W .Note that in this case necessarily m = dim k V = n . Remark 1.10.
Direct sums are stabilized by non-trivial one-parameter subgroups ofSL( V ). For example, the space L in (1.9) is fixed by any one-parameter subgroupacting with weight dim k W on U and weight − dim k U on W .Finally, we say that [ L ] ∈ Grass( n, Sym d V ) is decomposable if there is a non-trivialsubspace U ⊂ V and elements [ L ] ∈ Grass(dim k U, Sym d U ) and [ L ] ∈ Grass( n − dim k U, Sym d V ) such that L = L + L ⊂ Sym d V. Associated forms.
Next, we briefly recall the theory of associated forms asdeveloped in [5, 6, 2, 9]. To avoid trivialities, we adopt the following:
Assumption 1.11.
Assume n ≥
2, and that d ≥ n = 2, and d ≥ n ( d − ≥ d + 1.Let Grass( n, S d ) Res be the affine open subset in Grass( n, S d ) parameterizing lin-ear subspaces h g , . . . , g n i ⊂ S d such that g , . . . , g n form a regular sequence in S ,or, equivalently, such that the ideal ( g , . . . , g n ) is a complete intersection, or, equiv-alently, such that the resultant Res( g , . . . , g n ) is nonzero. Note that, if char( k ) ∤ ( d + 1)!, then f ∈ S d +1 is smooth if and only if ∇ f ∈ Grass( n, S d ) Res .For every U = h g , . . . , g n i ∈ Grass( n, S d ) Res , the ideal I U = ( g , . . . , g n ) is acomplete intersection ideal, and the k -algebra S/I U is a graded Gorenstein Artinlocal ring with socle in degree n ( d − k ) ∤ ( n ( d − ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 7
Macaulay’s theorem, there exists a unique up to scaling form A ( U ) ∈ D n ( d − suchthat A ( U ) ⊥ = I U . The form A ( U ) is called the associated form of g , . . . , g n by Alper and Isaev, whosystematically studied it in [2, Section 2]. In particular, they showed that the as-signment U → A ( U ) ∈ P D n ( d − gives rise to an SL( n )-equivariant associated formmorphism A : Grass( n, S d ) Res → P D n ( d − . When U = ∇ f for a smooth form f ∈ S d +1 , we set A ( f ) := A ( ∇ f ) = A ( h ∂f /∂x , . . . , ∂f /∂x n i ) , and, following Eastwood and Isaev [5], call A ( f ) the associated form of f . The definingproperty of A ( f ) is that the apolar ideal of A ( f ) is the Jacobian ideal of f : A ( f ) ⊥ = J f . This means that A ( f ) is a homogeneous Macaulay inverse system of the Milnor algebra M f = S/J f .Summarizing, when char( k ) ∤ ( n ( d − n )-equivariant morphisms: (cid:0) P S d +1 (cid:1) ∆ ∇ / / A & & ▼▼▼▼▼▼▼▼▼▼ Grass (cid:0) n, S d (cid:1) Res A w w ♥♥♥♥♥♥♥♥♥♥♥♥ P (cid:0) D n ( d − (cid:1) Remark 1.12.
In [2], Alper and Isaev define the associated form A ( g , . . . , g n ) as anelement of D n ( d − , which they achieve by choosing a canonical generator of the socleof S/ ( g , . . . , g n ) given by the Jacobian determinant of g , . . . , g n . For our purposes,it will suffice to consider A ( h g , . . . , g n i ) defined up to a scalar.2. First properties of direct sums and LDS forms
We begin by recording several immediate properties of direct sums and, more gen-erally, of LDS forms.
Lemma 2.1.
Suppose f ∈ S d +1 is a direct sum. Let ¯ f be its image in P S d +1 . Thenthe following hold: (1) If k = F , there is a non-trivial one-parameter subgroup ρ : G m ֒ → SL( V ) de-fined over k such that ρ ·h∇ f i = h∇ f i but ρ · f = f . Consequently, Stab
SL( V ) ( ¯ f ) is a proper subgroup of Stab
SL( V ) ( h∇ f i ) . (2) The set of k -points of { ¯ g ∈ P S d +1 | h∇ g i = h∇ f i} contains k ∗ = k \ { } . Although given over C , their proof applies whenever char( k ) = 0 or char( k ) > n ( d − MAKSYM FEDORCHUK (3)
If furthermore dim k h∇ f i = n , then ∇ f ∈ Grass( n, S d ) is a balanced directsum (cf. § k -points of { ¯ g ∈ P S d +1 | h∇ g i = h∇ f i and h∇ ( f − cg ) i 6 = 0 for all c ∈ k } contains k \ { , } .Proof. Suppose f = f ( x , . . . , x a ) + f ( x a +1 , . . . , x n )in some basis x , . . . , x n of V , where f , f = 0. Let ρ be the one-parameter subgroupof SL( V ) acting with weight ( n − a ) / gcd( n − a, a ) on the variables { x i } ai =1 and weight − a/ gcd( n − a, a ) on { x i } ni = a +1 . Since the weights are of opposite sign and coprime, ρ is a non-trivial subgroup of SL( V ) defined over k . Then we have(2.2) h∇ f i = h∇ f i ⊕ h∇ f i , where h∇ f i ⊂ k [ x , . . . , x a ] and h∇ f i ⊂ k [ x a +1 , . . . , x n ].It is clear that h∇ f i is invariant under ρ , and that f is not ρ -invariant because f , f = 0. This proves (1).Clearly, the forms f λ := f + λf , where λ ∈ k ∗ , all satisfy h∇ f λ i = h∇ f i , and are pairwise non-proportional for distinct values of λ . This proves (2).Suppose further that dim k h∇ f i = n so that f is concise. Then in (2.2), we havenecessarily that dim k h∇ f i = a and dim k h∇ f i = n − a . Thus ∇ f is a balanced directsum. We also see that ∇ ( f − c ( f + λf )) = (1 − c ) ∇ f + ( c − λ ) ∇ f = 0for all λ = 1. This proves (3). (cid:3) Under further assumptions on the characteristic of the field k , the converse toLemma 2.1(3) holds, and we have the following simple characterization of direct sumsin terms of their gradient points. Lemma 2.3.
Assume char( k ) ∤ ( d + 1)! and f ∈ S d +1 is a concise form (equivalently, dim k h∇ f i = dim k V = n ). Then f is a direct sum if and only if ∇ f ∈ Grass( n, S d ) is a direct sum if and only if ∇ f ∈ Grass( n, S d ) is a balanced direct sum.Proof. Suppose f = f ( x , . . . , x a ) + f ( x a +1 , . . . , x n ) in some basis x , . . . , x n of V ,where f , f = 0. Then using dim k h∇ f i = n , we deduce from (2.2) that h∇ f i is abalanced direct sum.Suppose ∇ f decomposes as a direct sum in a basis x , . . . , x n of V . Then ∇ f = h s , . . . , s b , t , . . . , t n − b i , for some s , . . . , s b ∈ k [ x , . . . , x a ] d and t , . . . , t n − b ∈ k [ x a +1 , . . . , x n ] d . ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 9
Then for every 1 ≤ i ≤ a and a + 1 ≤ j ≤ n , we can write ∂f /∂x i = u + v ,∂f /∂x j = u + v , where u , u ∈ k [ x , . . . , x a ] d and v , v ∈ k [ x a +1 , . . . , x n ] d . It follows that ∂ f∂x i ∂x j = ∂v ∂x j ∈ k [ x a +1 , . . . , x n ] d − . and ∂ f∂x j ∂x i = ∂u ∂x i ∈ k [ x , . . . , x a ] d − . In other words, for every 1 ≤ i ≤ a and a + 1 ≤ j ≤ n , we have ∂ f∂x i ∂x j ∈ k [ x , . . . , x a ] d − ∩ k [ x a +1 , . . . , x n ] d − = (0) . It follows that ∂ f∂x i ∂x j = 0 for all 1 ≤ i ≤ a and a + 1 ≤ j ≤ n . Using the assumptionon char( k ), we conclude that f ∈ k [ x , . . . , x a ] d +1 ⊕ k [ x a +1 , . . . , x n ] d +1 , and so f is a direct sum, in the same basis as ∇ f . (cid:3) For LDS forms, we have the following analog of Lemma 2.1.
Lemma 2.4.
Suppose f ∈ S d +1 is an LDS form. Let ¯ f be its image in P S d +1 . Thenthe following hold: (1) If k = F , there is a non-trivial one-parameter subgroup ρ : G m ֒ → SL( V ) suchthat ρ · h∇ f i = h∇ f i but ρ · f = f . Consequently, Stab
SL( V ) ( ¯ f ) is a propersubgroup of Stab
SL( V ) ( h∇ f i ) . (2) The set of k -points of { ¯ g ∈ P S d +1 | h∇ g i = h∇ f i} contains k ∗ . (3) If furthermore dim k h∇ f i = n , then ∇ f ∈ Grass( n, S d ) is decomposable.Proof. Suppose(2.5) f = a X i =1 x i ∂H ( x a +1 , . . . , x a ) ∂x a + i + G ( x a +1 , . . . , x n ) , where H and G are non-zero degree d + 1 forms, in a and n − a variables, respectively.Let ρ be the one-parameter subgroup of SL( V ) acting with weight ( n − a ) / gcd( n − a, a )on the variables { x i } ai =1 and weight − a/ gcd( n − a, a ) on { x i } ni = a +1 . Since the weightsare of opposite sign and coprime, ρ is a non-trivial subgroup of SL( V ) defined over k .The proof of (1) and (2) now proceeds as in the proof of Lemma 2.1 by noting that h∇ f i does not change if we multiply H by an element of k ∗ . Finally (3) follows from the fact that h∇ H i = h ∂f /∂x , . . . , ∂f /∂x a i ⊂ h∇ f i and h∇ H i ⊂ k [ x a +1 , . . . , x a ] . (cid:3) Direct sums and non-injectivity of ∇ Throughout this section, we work under:
Assumption 3.1.
The ground field k is algebraically closed and char( k ) ∤ ( d + 1)!.We explore the relationship between direct sum decomposability of a concise form f ∈ S d +1 (that is, the form defining a hypersurface which is not a cone in P n − ) andthe non-injectivity of the gradient morphism (see Definition 1.2) ∇ : P ( S d +1 ) c → Grass( n, S d )at the point ¯ f ∈ P ( S d +1 ). Our main result is the following complete characterization ofconcise forms that are uniquely determined by their gradient points, or, equivalently,by their Jacobian ideals. Theorem 3.2. (A) Suppose f ∈ S d +1 is a concise form. Then the following areequivalent: (1) f is either a direct sum or an LDS form. (2) The morphism ∇ is not injective at ¯ f ; that is, there exists ¯ g = ¯ f ∈ P ( S d +1 ) c such that ∇ f = ∇ g . (3) The morphism ∇ has positive fiber dimension at ¯ f ; that is, ∇ − ( ∇ ¯ f ) hasdimension ≥ at ¯ f .(B) In particular, if f ∈ S d +1 is a GIT semistable form with respect to the standard SL( n ) -action, then the following are equivalent: (1) f is a direct sum. (2) The morphism ∇ is not injective at ¯ f . (3) The morphism ∇ has positive fiber dimension at ¯ f . Partial results along these lines were known earlier. In a 1983 paper [3, § ∇ fails to be injectiveat ¯ f , then either f is a direct sum or f has a point of multiplicity deg( f ) −
1. Notethat all LDS forms of degree d + 1 have a point of multiplicity d and are GIT non-semistable by Lemma 1.7, but not all degree d + 1 hypersurfaces with multiplicity d points are GIT non-semistable. This shows that Wang’s result is slightly weaker thanour Theorem 3.2. In fact, some applications (such as, for example, in [10]) requireour stronger formulation.The key to our proof of Theorem 3.2 is Proposition 3.3, which is based on an ideaof Benson appearing in [3, Proposition 4.1] that was independently discovered andcommunicated to me by Alexander Isaev. We note that Wang’s result is proved by ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 11 the same method. We obtain our sharp result by pushing Benson’s method to its fulllogical conclusion.
Proof of Theorem 3.2.
We note that Part (B) follows immediately from Part (A) andLemma 1.7. We proceed to prove Part (A).Since ∇ f = ∇ g implies that ∇ ( λf + µg ) = ∇ f for a general [ λ : µ ] ∈ P k , we have(2) ⇒ (3), while the reverse implication is obvious. That (1) ⇒ (2) when f is a direct sum (respectively, a concise LDS form) followsfrom Lemma 2.1(2) (respectively, Lemma 2.4(2)).At last, we prove the implication (2) ⇒ (1) in Proposition 3.3 below, thus finishingthe proof of the theorem. (cid:3) The following slightly more general result does not require the conciseness assump-tion.
Proposition 3.3.
Keep Assumption 3.1. Suppose f, g ∈ S d +1 are arbitrary nonzeroforms that satisfy (3.4) h∇ g i ⊂ h∇ f i , or, equivalently, J g ⊂ J f . Then either g = λf , where λ ∈ k , or f is LDS, or f is adirect sum. Remark 3.5.
We regard any non-concise form as an instance of an LDS form obtainedby setting H = 0 in (1.5). Proof.
Following Benson [3, § x , . . . , x n of V suchthat(3.6) ∂g∂x ...∂g∂x n = M ∂f∂x ...∂f∂x n , where M is an n × n matrix in the Jordan normal form. If M = λ I n , where λ ∈ k , is ascalar multiple of the identity matrix, then by the Euler’s formula and the assumptionthat char( k ) ∤ ( d + 1)!, we have that g = λf , and we are done.It remains to consider two cases: either M has a Jordan block of size >
1, or M isdiagonal with at least two distinct eigenvalues. Claim 3.7. If M is not diagonal, then f is LDS.Proof. Let λ be any eigenvalue of M with a Jordan block of size greater than 1. Weassume that J , . . . , J ℓ are the Jordan blocks with eigenvalue λ of sizes m , . . . , m ℓ ≥ J ℓ +1 , . . . , J r , of sizes m ℓ +1 , . . . , m r , respectively, haveeither size 1 or eigenvalue different from λ . Note however that over non-algebraically closed fields, ∇ can be injective on k -points, but stillhave positive fiber dimension at some of them. An example is given by f = x d +1 + y d +1 ∈ F [ x, y ] d +1 . Double-index the variables so that { x , . . . , x n } = { x j , . . . , x jm j } rj =1 where { x j , . . . , x jm j } correspond to the Jordan block J j of size m j .For each j = 1 , . . . , ℓ , we suppose that J j = λ · · · λ · · · ... . . . . . . ... · · · · · · λ . Using (3.6), we then have( E j ) ∂g∂x j = λ ∂f∂x j ,∂g∂x j = λ ∂f∂x j + ∂f∂x j ,...∂g∂x jm j = λ ∂f∂x jm j + ∂f∂x jm j − . After passing to the second partials, and some elementary algebraic manipulations,we see that ∂ f∂x j ∂x j ′ t = 0 , for all 1 ≤ j, j ′ ≤ ℓ , and t = 1 , . . . , m j ′ − , (3.8) ∂ f∂x j ∂x j ′ t = 0 , for all 1 ≤ j ≤ ℓ < j ′ ≤ r and all t = 1 , . . . , m j ′ , (3.9) ∂ f∂x j ∂x j ′ m j ′ = ∂ f∂x j ∂x j ′′ m j ′′ , for all 1 ≤ j, j ′ , j ′′ ≤ ℓ .(3.10)Equation (3.8) with t = 1 implies that for all 1 ≤ j, j ′ ≤ ℓ , we have ∂ f∂x j ∂x j ′ = 0 . It follows that f is linear in the variables x , . . . , x ℓ , and so we can write f = x f + x f + · · · + x ℓ f ℓ + G, where f , . . . , f ℓ , G do not involve x , . . . , x ℓ . Moreover, we have that f j = ∂f∂x j , j = 1 , . . . , ℓ. Equations (3.8) and (3.9) now imply that f , . . . , f ℓ ∈ k [ x jm j ] ℓj =1 . ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 13
Next, (3.10) gives for all 1 ≤ j, j ′ ≤ ℓ that ∂f j ∂x j ′ m j ′ = ∂f j ′ ∂x jm j . It follows that there exists H ∈ k [ x jm j ] ℓj =1 such that f j = ∂H∂x jm j . Re-indexing the variables so that x j = x j and x ℓ + j = x jm j j = 1 , . . . , ℓ , we at lastsee that f = ℓ X j =1 x j ∂H ( x ℓ +1 , . . . , x ℓ ) ∂x j + ℓ + G ( x ℓ +1 , . . . , x n )and so f is LDS. (cid:3) Claim 3.11.
Suppose M is diagonal. Then either f is non-concise (and hence LDS)or f is a direct sum.Proof. Let λ be an eigenvalue of M , and x , . . . , x a are variables such that ∂g∂x i = λ ∂f∂x i , i = 1 , . . . , a and ∂g∂x j = λ j ∂f∂x j , j = a + 1 , . . . , n, where λ j = λ for j > a . Then ∂ f∂x i x j = 0 for all i ≤ a and j ≥ a + 1. Hence f = f ( x , . . . , x a ) + f ( x a +1 , . . . , x n ). (cid:3) This finishes the proof of proposition. (cid:3)
We obtain a number of consequences of Theorem 3.2 and Proposition 3.3, that inparticular describe the fibers of the gradient morphism. To state these results, weneed one more piece of terminology. Recall from [4, § f ∈ S d +1 , a decomposition f = f + · · · + f r , is called a maximally fine direct sum decomposition if V = E ( f ) ⊕ · · · ⊕ E ( f r ), and f i is not a direct sum in Sym d +1 E ( f i ), for all i = 1 , . . . , r . Corollary 3.12 (Fibers of ∇ ) . Suppose f ∈ S d +1 is not an LDS form and f = f + · · · + f r is a maximally fine direct sum decomposition. Suppose g ∈ S d +1 is suchthat h∇ g i ⊂ h∇ f i . Then g ∈ h f , . . . , f r i . In particular, for the gradient morphism ∇ : P ( S d +1 ) c → Grass( n, S d ) , we have (3.13) ∇ − ( ∇ f ) = { λ f + λ f + · · · + λ r f r | λ i ∈ k ∗ } . Proof.
Suppose ∇ g ⊂ ∇ f . Let x , . . . , x n be a basis of V adapted to the direct sumdecomposition E ( f ) = E ( f ) ⊕ · · · ⊕ E ( f r ). Take x i ∈ E ( f a ) and x j ∈ E ( f b ), where a = b . Then we have that ∂ g∂x i ∂x j ∈ Sym d E ( f a ) ∩ Sym d E ( f b ) = (0) , (cf. the proof of Lemma 2.3). It follows that the partials ∂ g∂x i ∂x j vanish whenever x i ∈ E ( f a ) and x j ∈ E ( f b ), where a = b . This implies that we can write g = g + · · · + g r , where E ( g i ) = E ( f i ). Moreover, from h∇ g i ⊂ h∇ f i , it follows that h∇ g i i ⊂ h∇ f i i .Applying Proposition 3.3, and the fact that each f i is not an LDS form, we concludethat g i is a scalar multiple of f i . The claim follows. (cid:3) Corollary 3.14.
Let ∇ : P ( S d +1 ) c → Grass( n, S d ) be the gradient morphism. Thenthe non-injectivity locus of ∇ is equal to the union of the direct sum locus and thelocus of LDS forms: n ¯ f ∈ P ( S d +1 ) c | there exists ¯ g = ¯ f ∈ P ( S d +1 ) nd such that ∇ f = ∇ g o = (cid:8) ¯ f ∈ P ( S d +1 ) c | either f is a direct sum or f is an LDS form (cid:9) . For concise forms of degree d + 1 ≥
3, Kleppe has established that a maximallyfine direct sum decomposition is unique [12, Theorem 3.7]. We obtain the followinggeneralization of his result:
Corollary 3.15.
Suppose f is not an LDS form. Then a maximally fine direct sumdecomposition of f is unique.Proof. Suppose f = f + · · · + f s = g + · · · + g t are two maximally fine direct sumdecompositions. Then by Corollary 3.12, we have that { λ f + λ f + · · · + λ r f r | λ i ∈ k ∗ } = ∇ − ( ∇ f ) = { λ g + λ g + · · · + λ t g t | λ i ∈ k ∗ } . It follows that g i ∈ h f , . . . , f r i . Since g i is not a direct sum itself, we must have g i = c i f j for some j = 1 , . . . , s and some c i ∈ k ∗ . It is then immediate that t = s and c i = 1 for all i = 1 , . . . , s . (cid:3) Connection with the work of Buczy´nska-Buczy´nski-Kleppe-Teitler.
The follow-ing result is established in [4] under the assumption that k is an algebraically closedfield of characteristic 0: Theorem 3.16 (see [4, Theorem 1.7]) . A concise form f ∈ S d +1 is either a directsum or an LDS form if and only if the apolar ideal f ⊥ has a minimal generator indegree d + 1 . Since LDS forms are GIT non-semistable and in particular singular, this translatesinto a computable and effective criterion for recognizing whether a smooth form f isa direct sum over an algebraically closed field. ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 15
The following simple observation reconciles the above result with Theorem 3.2.
Proposition 3.17.
Keep Assumption 3.1. Consider the apolarity action
D × S → S given by F ( z , . . . , z n ) ◦ g = F (cid:18) ∂∂x , . . . , ∂∂x n (cid:19) g ( x , . . . , x n ) . Then for a form f ∈ S d +1 , the apolar ideal f ⊥ = { F ∈ D | F ◦ f = 0 } has a minimal generator in degree d + 1 if and only if there exists g ∈ S d +1 , not ascalar multiple of f , such that ∇ f ⊂ ∇ g .Proof. Let I = f ⊥ ⊂ D . Then S/I is a Gorenstein Artin k -algebra with socle indegree d + 1. By definition, I has a minimal generator in degree d + 1 if and only ifand dim k ( I d +1 / D I d ) ≥
1, or equivalently if D I d ( I d +1 .On the other hand, D I d ( I d +1 if and only if { g ∈ S d +1 | F ◦ g = 0 , for all F ∈ I d +1 } ( { g ∈ S d +1 | F ◦ g = 0 for all F ∈ D I d } . We now compute that { g ∈ S d +1 | F ◦ g = 0 , for all F ∈ I d +1 } = h f i , and { g ∈ S d +1 | F ◦ g = 0 for all F ∈ D I d } = { g ∈ S d +1 | F ◦ ∂g∂x i = 0 for all F ∈ I d , and all i = 1 , . . . , n } = { g ∈ S d +1 | F ◦ ∂g∂x i = 0 for all F ∈ ( f ⊥ ) d , and all i = 1 , . . . , n } = { g ∈ S d +1 | ∇ f ⊂ ∇ g } , where we have used the equality I d = ( f ⊥ ) d = { F ∈ D d | F ◦ ∂f /∂x i = 0 for all i = 1 , . . . , n } . We conclude that D I d ( I d +1 if and only if h f i ( { g ∈ S d +1 | ∇ f ⊂ ∇ g } , which is precisely the condition that there exists g ∈ S d +1 , not a scalar multiple of f ,such that ∇ f ⊂ ∇ g . (cid:3) Direct sum decomposability of smooth forms
We keep Assumption 1.11.
Theorem 4.1.
Suppose k is an arbitrary field satisfying char( k ) ∤ ( n ( d − . Let f ∈ S d +1 be a smooth form. Then the following are equivalent: (1) f is a direct sum. (2) ∇ f is a balanced direct sum. (3) A ( f ) is a balanced direct product. (4) A ( f ) is a direct product. (5) ∇ f admits a non-trivial G m -action defined over k . (6) A ( f ) admits a non-trivial G m -action defined over k .Moreover, if z , . . . , z n is a basis of V ∨ in which A ( f ) factors as A ( f ) = G ( z , . . . , z a ) G ( z a +1 , . . . , z n ) , then f decomposes as f = f ( x , . . . , x a ) + f ( x a +1 , . . . , x n ) in the dual basis x , . . . , x n of V .Proof of Theorem 4.1. The implications (1) = ⇒ (2) and (1) = ⇒ (5) are in Lemma2.1. The implication (2) = ⇒ (1) is in Lemma 2.3.The equivalence (2) ⇐⇒ (3) is proved in Proposition 4.2 below. This concludesthe proof of equivalence for the first three conditions.Next we prove (4) = ⇒ (3). Suppose A ( f ) = G ( z , . . . , z a ) G ( z a +1 , . . . , z n ) is adirect product decomposition in a basis z , . . . , z n of V ∨ . Let x , . . . , x n be the dualbasis of V . Suppose x d · · · x d n n is the smallest with respect to the graded reverselexicographic order monomial of degree n ( d −
1) that does not lie in ( J f ) n ( d − . Since z d · · · z d n n must appear with a nonzero coefficient in A ( f ), we have that d + · · · + d a = deg G . On the other hand, by [9, Lemma 4.1], we have that d + · · · + d a ≤ a ( d − G ≤ a ( d − G ≤ ( n − a )( d − A ( f ) = G G is a balanceddirect product decomposition. Alternatively, we can consider a diagonal action of G m ⊂ SL( V ) on V that acts on V ∨ as follows: t · ( z , . . . , z n ) = (cid:16) t ( n − a ) z , . . . , t ( n − a ) z a , t − a z a +1 , . . . , t − a z n (cid:17) . Then A ( f ) is homogeneous with respect to this action, and has weight ( n − a ) deg G − a deg G . However, the relevant parts of the proof of [8, Theorem 1.2] go through toshow that A ( f ) satisfies the Hilbert-Mumford numerical criterion for semistability.This forces ( n − a ) deg G − a deg G = 0 . We now turn to the last two conditions. First, the morphism A is an SL( n )-equivariant locally closed immersion by [2, § ⇐⇒ (6). The implication (5) = ⇒ (1) follows from theproof of [8, Theorem 1.0.1] that shows that for a smooth f , the gradient point ∇ f hasa non-trivial G m -action if and only if f is a direct sum. We note that even thoughstated over C , the relevant parts of the proof of [8, Theorem 1.0.1] use only [8, Lemma3.5], which remains valid over a field k with char( k ) = 0 or char( k ) > d + 1, and thefact that a smooth form over any field must satisfy the Hilbert-Mumford numericalcriterion for stability. (cid:3) Proposition 4.2.
Let d ≥ . Suppose k is a field with char( k ) = 0 or char( k ) >n ( d − . Then an element U ∈ Grass( n, Sym d V ) Res is a balanced direct sum if and
ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 17 only if A ( U ) is a balanced direct product. Moreover, if z , . . . , z n is a basis of V ∨ inwhich A ( U ) factors as a balanced direct product, then U decomposes as a balanceddirect sum in the dual basis x , . . . , x n of V .Proof. The forward implication is an easy observation. Consider a balanced directsum U = h g , . . . , g n i ∈ Grass( n, k [ x , . . . , x n ] d ) Res , where g , . . . , g a ∈ k [ x , . . . , x a ] d and g a +1 , . . . , g n ∈ k [ x a +1 , . . . , x n ] d . Then, up to a nonzero scalar, A ( U ) = A ( g , . . . , g a ) A ( g a +1 , . . . , g n ) , where A ( g , . . . , g a ) ∈ k [ z , . . . , z a ] a ( d − and A ( g a +1 , . . . , g n ) ∈ k [ z a +1 , . . . , z n ] ( n − a )( d − ;see [9, Lemma 2.11], which also follows from the fact that on the level of algebras, wehave k [ x , . . . , x n ]( g , . . . , g n ) ≃ k [ x , . . . , x a ]( g , . . . , g a ) ⊗ k k [ x a +1 , . . . , x n ]( g a +1 , . . . , g n ) . Suppose now A ( U ) is a balanced direct product in a basis z , . . . , z n of V ∨ :(4.3) A ( U ) = F ( z , . . . , z a ) F ( z a +1 , . . . , z n ) , where deg( F ) = a ( d −
1) and deg( F ) = ( n − a )( d − x , . . . , x n be the dualbasis of V , and let I U ⊂ k [ x , . . . , x n ] be the complete intersection ideal spanned bythe elements of U . We have that I U = A ( U ) ⊥ ⊂ k [ x , . . . , x n ] . It is then evident from (4.3) and the definition of an apolar ideal that(4.4) ( x , . . . , x a ) a ( d − ⊂ I U and(4.5) ( x a +1 , . . . , x n ) ( n − a )( d − ⊂ I U . We also have the following observation:
Claim 4.6. dim k (cid:0) U ∩ ( x , . . . , x a ) (cid:1) = a, dim k (cid:0) U ∩ ( x a +1 , . . . , x n ) (cid:1) = n − a. Proof.
By symmetry, it suffices to prove the second statement. Since U is spanned bya length n regular sequence of degree d forms, we have that dim k (cid:0) U ∩ ( x a +1 , . . . , x n ) (cid:1) ≤ n − a . Suppose we have a strict inequality. Let R := k [ x , . . . , x n ] / ( I U , x a +1 , . . . , x n ) ≃ k [ x , . . . , x a ] /I ′ . Then I ′ is generated in degree d , and has at least a + 1 minimal generators in thatdegree. It follows that the top degree of R is strictly less than a ( d − I ′ a ( d − = k [ x , . . . , x a ] a ( d − (cf. [9, Lemma 2.7]). But then k [ x , . . . , x a ] a ( d − ⊂ ( x a +1 , . . . , x n ) + I U . Using (4.5), this gives k [ x , . . . , x a ] a ( d − k [ x a +1 , . . . , x n ] ( n − a )( d − ⊂ I U . Thus every monomial of k [ z , . . . , z a ] a ( d − k [ z a +1 , . . . , z n ] ( n − a )( d − appears with coefficient 0 in A ( U ), which contradicts (4.3). (cid:3) At this point, we can apply [9, Proposition 3.1] to conclude that U ∩ k [ x , . . . , x a ] d contains a regular sequence of length a and that U ∩ k [ x a +1 , . . . , x n ] d contains a regularsequence of length n − a . This shows that U decomposes as a balanced direct sum inthe basis x , . . . , x n of V . However, for the sake of self-containedness, we proceed togive a more direct argument:By Claim (4.6), there exists a regular sequence s , . . . , s a ∈ k [ x , . . . , x a ] d such that( s , . . . , s a ) = (cid:0) g ( x , . . . , x a , , . . . , , . . . , g n ( x , . . . , x a , , . . . , (cid:1) and a regular sequence t , . . . , t n − a ∈ k [ x a +1 , . . . , x n ] d such that( t , . . . , t n − a ) = (cid:0) g (0 , . . . , , x a +1 , . . . , x n ) , . . . , g n (0 , . . . , , x a +1 , . . . , x n ) (cid:1) . Let W := h s , . . . , s a , t , . . . , t n − a i ∈ Grass( n, S d ) Res and let I W be the ideal generated by W . We are going to prove that U = W , whichwill conclude the proof of the proposition.Since char( k ) = 0 or char( k ) > n ( d − U = W , we need to show that the ideals I U and I W coincide in degree n ( d − I W ) n ( d − ⊂ ( I U ) n ( d − .Since s , . . . , s a is a regular sequence in k [ x , . . . , x a ] d , we have that k [ x , . . . , x a ] a ( d − ⊂ ( s , . . . , s a ) . Similarly, we have that k [ x a +1 , . . . , x n ] ( n − a )( d − ⊂ ( t , . . . , t n − a ) . Together with (4.4) and (4.5), this gives(4.7) ( x , . . . , x a ) a ( d − + ( x a +1 , . . . , x n ) ( n − a )( d − ⊂ I U ∩ I W . Set J := ( x , . . . , x a ) a ( d − + ( x a +1 , . . . , x n ) ( n − a )( d − . It remains to show that( I W ) n ( d − ⊂ ( I U ) n ( d − + J n ( d − . To this end, consider a X i =1 q i s i + n − a X j =1 r j t j ∈ ( I W ) n ( d − , where q , . . . , q a , r , . . . , r n − a ∈ S n ( d − − d . Since s , . . . , s a ∈ k [ x , . . . , x a ] d , and weare working modulo J , we can assume that q i ∈ ( x a +1 , . . . , x n ) ( n − a )( d − , for all i =1 , . . . , a . Similarly, we can assume that r j ∈ ( x , . . . , x a ) a ( d − , for all j = 1 , . . . , n − a . ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 19
By construction, we have s , . . . , s a ∈ I U + ( x a +1 , . . . , x n ) and t , . . . , t n − a ∈ I U +( x , . . . , x a ). Using this, and (4.7), we conclude that a X i =1 q i s i + n − a X j =1 r j t j ∈ I U + J. This finishes the proof of the proposition. (cid:3)
We use Theorem 4.1 to give an alternate proof of Corollary 3.15 for smooth forms,deducing it from the fact that a polynomial ring over a field is a UFD:
Proposition 4.8.
Keep Assumption 1.11. Suppose f ∈ S d +1 is a smooth form. Then f has a unique maximally fine direct sum decomposition.Proof of Proposition 4.8. Suppose f = f + · · · + f s = g + · · · + g t are two maximallyfine direct sum decompositions. Then A ( f ) · · · A ( f s ) = A ( g ) · · · A ( g t ) ∈ P D n ( d − , where V ∨ = ⊕ si =1 E ( A ( f i )) = ⊕ tj =1 E ( A ( g j )). Suppose some A ( f i ) shares irreduciblefactors with more than one A ( g j ). Then by the uniqueness of factorization in D , wemust have a non-trivial factorization A ( f i ) = G G such that E ( G ) ∩ E ( G ) = (0).Then A ( f i ) is a direct product, and so f i must be a direct sum by Theorem 4.1,contradicting the maximality assumption. Therefore, no A ( f i ) shares an irreduciblefactor with more than one A ( g j ); and, by symmetry, no A ( g j ) shares an irreduciblefactor with more than one A ( f i ). It follows that s = t and, up to reordering, A ( f i ) = A ( g i ), and thus E ( A ( f i )) = E ( A ( g i )), for all i = 1 , . . . , t . We conclude that E ( f i ) = E ( g i ), which using f + · · · + f t = g + · · · + g t forces f i = g i , for all i = 1 , . . . , t . (cid:3) Necessary conditions for direct sum decomposability
Our next two results give easily verifiable necessary conditions for an arbitraryform to be a direct sum. They hold over a large enough field, with no restrictionon characteristic, and are independent of the results of Sections 3 and 4. We applythem to prove that determinant and pfaffian-like polynomials are not direct sums inCorollaries 5.7 and 5.8. We keep notation of § Theorem 5.1.
Suppose f is a form in S = k [ x , . . . , x n ] d and that card( k ) ≫ d . (1) Let b = dim k ∇ f . If f has a factor g such that dim k ∇ g ≤ ⌊ b − ⌋ , then f isnot a direct sum. (2) If f has a repeated factor, then f is not a direct sum. Remark 5.2.
It is possible to pinpoint precisely how large the cardinality of k hasto be, but since we do not have an application for this, we will not do so. Corollary 5.3.
Suppose f is a form with dim k ∇ f ≥ , and that card( k ) ≫ deg( f ) .If f has a linear factor, then f is not a direct sum. The result of this corollary was proved in [4, Proposition 2.12] using a criterionof Smith and Stong [14] for indecomposability of Gorenstein Artin algebras into con-nected sums. Our proof of the linear factor case of Theorem 5.1 and the statementfor higher degree factors appear to be new.
Proof of Theorem 5.1.
We apply Lemma 2.1. For (1), suppose f = gh , and that insome basis of V we have f = f ( x , . . . , x a ) + f ( x a +1 , . . . , x n ) ∈ S, and that dim k ∇ f = b while dim k ∇ g ≤ ⌊ b − ⌋ . Let ρ be the 1-PS subgroup of SL( V )acting with weight w := n − a gcd( n − a,a ) on { x i } ai =1 and weight w := − a gcd( n − a,a ) on { x i } ni = a +1 . Then ρ fixes ∇ f , and h∇ f i = h∇ f i ( w d ) ⊕ h∇ f i ( w d ) is the decomposition into the ρ -weight-spaces. By the assumption on the cardinalityof k , these two weight subspaces are distinct.Since h∇ f i ⊂ g h∇ h i + h h∇ g i , we havedim k ( h∇ f i ∩ g h∇ h i ) ≥ b − dim k h h∇ g i = b − dim k h∇ g i ≥ (cid:24) b + 12 (cid:25) . It follows by dimension considerations that some nonzero multiple gr , where r ∈h∇ h i , belongs to one of the two weight-spaces h∇ f i ( w i d ) of ρ in ∇ f . Thus g itself ishomogeneous with respect to ρ . Again invoking the assumption on the cardinality of k , we conclude that either g ∈ k [ x , . . . , x a ] or g ∈ k [ x a +1 , . . . , x n ]. This forces either f = 0 or f = 0, respectively. A contradiction!For (2), suppose f is a direct sum with a repeated factor g . Let ρ be the 1-PSof SL( V ) as above. Since ∇ f ⊂ ( g ), some nonzero multiple gr , where r ∈ S d − deg g ,belongs to one of the two weight-spaces of ρ in ∇ f . It follows that g is homogeneouswith respect to ρ and so we obtain a contradiction as in (1). (cid:3) Our next result needs the following definition that is standard in the theory of GITstability of homogeneous forms:
Definition 5.4.
Given a basis x , . . . , x n of V and a nonzero f ∈ S d , we define thestate of f to be the set of multi-indices Ξ( f ) ⊂ { ( d , . . . , d n ) ∈ Z n ≥ | d + · · · + d n = d } such that f = X ( d ,...,d n ) ∈ Ξ( f ) a ( d ,...,d n ) x d · · · x d n n , where a ( d ,...,d n ) ∈ k ∗ .In other words, the state of f is the set of monomials appearing with nonzero coeffi-cient in f . We set Ξ(0) = ∅ . Theorem 5.5.
Suppose k = F . Suppose f ∈ S d , where d ≥ , is such that in somebasis x , . . . , x n of V the following conditions hold: (1) Ξ( ∂f /∂x i ) = ∅ for all ≤ i ≤ n . (2) Ξ( ∂f /∂x i ) ∩ Ξ( ∂f /∂x j ) = ∅ for all ≤ i < j ≤ n . ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 21 (3)Ξ( f ) = n ( d , . . . , d n ) (cid:12)(cid:12)(cid:12) char( k ) ∤ d i for some ≤ i ≤ n and Ξ ∂ ( x d · · · x d n n ) ∂x i ! ⊂ ∪ nj =1 Ξ( ∂f /∂x j ) , for all ≤ i ≤ n o . (4) The graph with the vertices in { , . . . , n } and the edges given by (cid:26) ( ij ) | ∂ f∂x i ∂x j = 0 (cid:27) is connected.Then f is not a direct sum. Remark 5.6.
In words, (2) says that no two first partials of f share a commonmonomial, and (3) says that any monomial all of whose nonzero first partials appearin first partials of f must appear in f .As an immediate corollary of this theorem, we show that the n × n generic deter-minant and permanent polynomials, and the 2 n × n generic pfaffian polynomials, aswell as any other polynomial of the same state, are not direct sums when n ≥ Corollary 5.7 (Determinant-like polynomials are not direct sums) . Let n ≥ . Sup-pose k = F . Suppose S = k [ x i,j ] ni,j =1 and f = P σ ∈ S n a σ x ,σ (1) · · · x n,σ ( n ) , where a σ ∈ k ∗ . Then f is not a direct sum. Corollary 5.8 (Pfaffian-like polynomials are not direct sums) . Let n ≥ . Suppose k = F . Suppose S = k [ x i,j ] ≤ i
Proof of Theorem 5.5.
If char( k ) = p , we setΞ p := { ( d , . . . , d n ) | p divides d i for all 1 ≤ i ≤ n } to be the set of all monomials whose gradient point is trivial.Suppose f is a direct sum. Note that Assumption (1) implies that dim k ∇ f = n .Then by Lemma 2.1(3), and the assumption that k \ { , } 6 = ∅ , there exists a form g such that ∇ g = ∇ f and ∇ ( g − cf ) = 0 for all c ∈ k . Since ∇ g = ∇ f , then byAssumption (3), we must have Ξ( g ) ⊂ Ξ( f ) ∪ Ξ p . Then Ξ( ∂g/∂x i ) ⊂ Ξ( ∂f /∂x i ). Since ∂g/∂x i ∈ h∇ f i , Assumption (2) implies that in fact ∂g∂x i = c i ∂f∂x i , for some c i ∈ k. Comparing the second partials, and using Assumption (4) we conclude that c i = c j for all 1 ≤ i < j ≤ n . We obtain ∇ ( g − c f ) = (0), which is a contradiction. (cid:3) Finding a balanced direct product decomposition algorithmically
In this section, we show how Theorem 4.1 reduces the problem of finding a directsum decomposition of a given smooth form f to a polynomial factorization problem.To begin, suppose that we are given a smooth form f ∈ Sym d +1 V in some basis of V . Then the associated form A ( f ) is computed in the dual basis of V ∨ as the formapolar to the Jacobian ideal J f . In fact, since we know that A ( f ) has degree n ( d − n ( d −
1) apolar to the space ( J f ) n ( d − .To apply Theorem 4.1, we now need to determine if A ( f ) ∈ Sym n ( d − V ∨ decom-poses as a balanced direct product, and if it does, then in what basis of V ∨ . Thefollowing simple lemma explains how to do it (cf. § Lemma 6.1.
Suppose char( k ) ∤ ( n ( d − . For a smooth f ∈ S d +1 , the associatedform A ( f ) is a balanced direct product if and only if there is a non-trivial factorization A ( f ) = G G such that (6.2) V = (cid:0) G ⊥ (cid:1) + (cid:0) G ⊥ (cid:1) , or, equivalently, E ( G ) ∩ E ( G ) = (0) ⊂ V ∨ . Moreover, in this case, we have (cid:0) G ⊥ (cid:1) ∩ (cid:0) G ⊥ (cid:1) = (0) ⊂ V and A ( f ) decomposes as abalanced direct product in any basis of V ∨ such that its dual basis is compatible withthe direct sum decomposition in Equation (6.2) .Proof. The equivalence of the two conditions in (6.2) follows from the fact that E ( G i ) ⊂ V ∨ is dual to ( G ⊥ i ) ⊂ V . The claim now follows from definitions andTheorem 4.1 by observing that for any non-trivial factorization A ( f ) = G G , wehave (cid:0) G ⊥ (cid:1) ∩ (cid:0) G ⊥ (cid:1) ⊂ (cid:0) A ( f ) ⊥ (cid:1) = ( J f ) = (0). (cid:3) An algorithm for direct sum decompositions.
Suppose k is a field, with ei-ther char( k ) = 0 or char( k ) > max { n ( d − , d +1 } , for which there exists a polynomialfactorization algorithm. Let f ∈ k [ x , . . . , x n ] d +1 , where d ≥ Step 1:
Compute J f = ( ∂f /∂x , . . . , ∂f /∂x n ) up to degree n ( d −
1) + 1. If( J f ) n ( d − = k [ x , . . . , x n ] n ( d − , then f is not smooth and we stop; otherwise, continue. We recall that we call a homogeneous form f ∈ k [ x , . . . , x n ] smooth if it defines a smoothhypersurface in P n − . ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 23
Step 2:
Compute A ( f ) as the dual to ( J f ) n ( d − : h A ( f ) i = (cid:8) T ∈ k [ z , . . . , z n ] n ( d − | g ◦ T = 0 , for all g ∈ ( J f ) n ( d − (cid:9) . This can be done as follows:Compute the degree n ( d −
1) part of the Gr¨obner basis of the Jacobian ideal J f ,say, using the graded reverse lexicographic order. Suppose { m j } Nj =1 , where N =dim k k [ x , . . . , x n ] n ( d − , are the monomials in k [ x , . . . , x n ] n ( d − in the given order.Then the output will be of the form( J f ) n ( d − = ( m − c m i , . . . , m i − − c i − m i , m i +1 , . . . , m N ) , where m i is the unique monomial of degree n ( d −
1) which is not an initial monomialof an element of J n ( d − . Let { b m j } Nj =1 be the dual monomials in k [ z , . . . , z n ]. Namely, m i ◦ b m j = ( , i = j, , i = j . For example, \ x d · · · x d n n = 1( d )! · · · ( d n )! z d · · · z d n n . Then A ( f ) = c m i + X j
Compute the irreducible factorization of A ( f ) in k [ z , . . . , z n ] and check forthe existence of balanced direct product factorizations using Lemma 6.1. If any exist,then f is a direct sum; otherwise, f is not a direct sum. Step 4:
For every balanced direct product factorization of A ( f ), Lemma 6.1 gives abasis of V in which f decomposes as a direct sum.The above algorithm was implemented in a Macaulay 2 package written by JustinKim and Zihao Fang (its source code is available upon request). In what follows, wegive a few examples of the algorithm in action. Remark 6.3.
Jaros law Buczy´nski has pointed out that already
Step 2 in the abovealgorithm is computationally highly expensive when n and d are large. However, it isreasonably fast when both n and d are small, with Example 6.6 below taking only afew seconds. Example 6.4.
Consider f = x + 3 x x + 3 x x + 2 x + 3 x x + 6 x x x + 4 x x +3 x x + 4 x x + 2 x in Q [ x , x , x ] from the introduction. Its discriminant is nonzeroand so we can compute the associated form of f . We have A ( f ) = − z + z z + 12 z z + z z − z z z + 12 z z = 12 z ( − z + z z + 12 z + z z − z z + 12 z ) . The first factor is a polynomial in z and the second factor is a polynomial in z − z and z − z . It follows that A ( f ) is a balanced direct product and so f is a direct sum. Indeed, the basis of V dual to the basis { z , z − z , z − z } of V ∨ is precisely { x + x + x , x , x } . In this basis, the original polynomial becomes a direct sum: f = ( x + x + x ) + x + x x + x x + x . Example 6.5 (Binary quartics) . Suppose k is an algebraically closed field of charac-teristic 0. Then every smooth binary quartic has a standard form f t = x + x + tx x , t = ± . Up to a scalar, the associated form of F t is A ( f t ) = t ( z + z ) − z z . Clearly, A ( f t ) is singular if and only if t = 0, or t = ±
6. For these values of t , A ( f t )is in fact a balanced direct product, and so f t is a direct sum. Namely, up to scalars,we have: A ( f ) = z z , f = x + x ,A ( f ) = ( z − z ) = ( z − z ) ( z + z ) , f = ( x − x ) + ( x + x ) ,A ( f − ) = ( z + z ) = ( z − iz ) ( z + iz ) , f − = ( x − ix ) + ( x + ix ) . Note that over R , the associated form A ( f − ) = ( z + z ) is not a balanced directproduct. Hence f − is not a direct sum over R by Theorem 4.1. Since the apolarideal of f − is the same over R and over C , this example illustrates that the directsum decomposability criterion of [4] fails over non-closed fields. Example 6.6.
Consider the following element in Q [ x , x , x , x ] : f = x + 4 x x + 6 x x + 4 x x + 2 x + 8 x x + 24 x x x + 24 x x x + 8 x x + 24 x x + 48 x x x + 24 x x + 32 x x + 32 x x + 17 x − x x − x x x − x x x − x x − x x x − x x x x − x x x − x x x − x x x − x x + 54 x x + 108 x x x + 54 x x + 216 x x x + 217 x x x + 216 x x − x x − x x − x x + 82 x . Then its associated form is A ( f ) = 9785 z − z z + 26370 z z − z z + 15 z z + 24 z z − z z + 40920 z z z − z z z − z z z + 60 z z z − z z z + 8730 z z − z z z + 6390 z z z + 30 z z z − z z − z z z + 15 z z z − z z z + 15 z z + 120 z z z + 12 z z − z z z − z z − z z z + 17820 z z z − z z z + 90 z z z − z z z + 21600 z z z z − z z z z + 6480 z z z − z z z z + 4320 z z z z − z z z + 90 z z z − z z + 7140 z z z + 2970 z z z − z z z + 15 z z z + 3120 z z z − z z z z − z z z z + 1080 z z z − z z z z + 720 z z z z − z z z + 15 z z z − z z + 2880 z z z + 1440 z z z − z z z z + 30 z z + 240 z z z + 120 z z z − z z z z + 36 z z + 2 z z . One checks that A ( f ) = G G , where G = z with E ( G ) = z , and E ( G ) = h z +2 z , z + z , z + z i , is a balanced direct product factorization of A ( f ). It follows that f is a direct sum. In fact, f is projectively equivalent to x + ( x + x + x + x x x ) . ECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 25
Acknowledgments
The author is grateful to Jarod Alper for an introduction to the subject, AlexanderIsaev for numerous stimulating discussions that inspired this work, and Zach Teitlerfor questions that motivated most of the results in Section 5. We thank ZhenjianWang for alerting us to his earlier work [17] related to Section 3 and interestingquestions.The author was partially supported by the NSA Young Investigator grant H98230-16-1-0061 and Alfred P. Sloan Research Fellowship. Justin Kim and Zihao Fang wrotea Macaulay 2 package for computing associated forms while supported by the BostonCollege Undergraduate Research Fellowship grant under the direction of the author.
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