Direct sum decomposability of polynomials and factorization of associated forms
For a homogeneous polynomial with a non-zero discriminant, we interpret direct sum decomposability of the polynomial in terms of factorization properties of the Macaulay inverse system of its Milnor algebra. This leads to an if-and-only-if criterion for direct sum decomposability of such a polynomial, and to an algorithm for computing direct sum decompositions over any field, either of characteristic or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. We also give simple necessary criteria for direct sum decomposability of arbitrary homogeneous polynomials over arbitrary fields and apply them to prove that many interesting classes of homogeneous polynomials are not direct sums.
A homogeneous polynomial is called a direct sum if, after a linear change of variables, it can be written as a sum of two or more polynomials in disjoint sets of variables:
When is a homogeneous polynomial over defining an isolated hypersurface singularity in , the geometric significance of such decomposition stems from the classical Sebastiani-Thom theorem  that describes the monodromy operator of the singularity as a tensor product of the monodromy operators of the singularities and . Direct sums are also the subject of a well-known symmetric Strassen’s additivity conjecture postulating that the Waring rank of in (1.1) is the sum of the Waring ranks of and (see, for example, ).
In this paper, we give a new criterion for recognizing when a smooth form
Recall that to a smooth form of degree in variables, one can assign a degree form in (dual) variables, called the associated form of ([1, 2, 4]). The associated form is defined as a Macaulay inverse system of the Milnor algebra of , which simply means that the apolar ideal of coincides with the Jacobian ideal of :
Such definition leads to an observation that for a smooth form that is written as a sum of two forms in disjoint sets of variables, the associated form decomposes as a product of two forms in disjoint sets of (dual) variables ([7, Lemma 2.11]). For example, up to a scalar,
The main purpose of this paper is to prove the converse statement, and thus establish an if-and-only-if criterion for direct sum decomposability of a smooth form in terms of the factorization properties of its associated form (see Theorem 1.6).
In Lemma 2.1, 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. It is then applied in Section 3 to prove that a wide class of homogeneous forms contains no direct sums. In Theorem 1.8, we show that this simple necessary condition is in fact sufficient when a form is GIT stable over an algebraically closed field.
1.1. Notation and conventions
Let be a field. The -linear span of a subset of a -vector space will be denoted by . If is a representation of the multiplicative group , then, for every , we denote by the weight-space of the action of weight .
Let be a vector space over with . We set , and . Homogeneous elements of and will be called forms. We have a differentiation action of on (also known as the apolar pairing). Namely, if is a basis of , and is the dual basis of , then the pairing
is given by
Given a homogeneous nonzero , the apolar ideal of is
and the space of essential variables of is
If or , then the pairing is perfect, and for every , we have (cf. [3, §2.2]). Furthermore, the graded -algebra is a Gorenstein Artin local ring with socle in degree , and a well-known theorem of Macaulay establishes a bijection between graded Gorenstein Artin quotients of socle degree and elements of (see, e.g., [8, Lemma 2.12] or [5, Exercise 21.7]).
Let be a basis of . The gradient point of is defined to be:
The Jacobian ideal of is , and the Milnor algebra of is .
Even though we allow to have positive characteristic, we do not take to be the divided power algebra (cf. [8, Appendix A]), as the reader might have anticipated. The reason for this is that at several places we cannot avoid but to impose a condition that is large enough (or zero). In this case, the divided power algebra is isomorphic to up to the needed degree.
For a homogeneous ideal , we denote by the closed subscheme of defined by . We say that a form is smooth if the hypersurface is smooth over (this is, of course, equivalent to being non-singular over the algebraic closure of ). The locus of smooth forms in will be denoted by .
1.2. Direct sums and products
Recall from  that is called a direct sum (or a form of Sebastiani-Thom type) if there is a direct sum decomposition and nonzero and such that . In other words, is a direct sum if and only if for some choice of a basis of , we have that
where , and . Recall also that is called degenerate if there exists such that .
By analogy with direct sums, we will call a nonzero form a direct product if there is a non-trivial direct sum decomposition and for some and . In other words, a nonzero homogeneous is a direct product if and only if for some choice of a basis of , we have that
Furthermore, we call a direct product decomposition in (1.3) balanced if
Note that a non-trivial factorization is a direct product decomposition if and only if .
Note that the roles of and are interchangeable in §1.1, and so for , we can define the apolar ideal and the space of essential variables . With this notation, if or , then is a direct sum if and only if we can write , where and . Furthermore, if or , we have that is dual to , and .
We say that is a balanced direct sum if there is a non-trivial direct sum decomposition and elements and such that
1.3. Associated forms
We briefly recall the theory of associated forms as developed in . Let be the affine open subset in parameterizing linear spaces such that form a regular sequence in . Note that, if or , then is smooth if and only if .
For every , the ideal is a complete intersection ideal, and the -algebra is a graded Gorenstein Artin local ring with socle in degree . Suppose or . Then, by Macaulay’s theorem, there exists a unique up to scaling form such that
The form is called the associated form of by
Alper and Isaev, who systematically studied it in [2, Section 2].
In particular, they showed
When for a smooth form , we set
and, following Eastwood and Isaev , call the associated form of . The defining property of is that
This means that is a Macaulay inverse system of the Milnor algebra .
Summarizing, when or , we have the following commutative diagram of -equivariant morphisms:
In , Alper and Isaev define the associated form as an element of , which they achieve by choosing a canonical generator of the socle of given by the Jacobian determinant of . For our purposes, it will suffice to consider defined up to a scalar.
1.4. Main results
Let . Suppose that either or . Let be a smooth form. Then the following are equivalent:
is a direct sum.
is a balanced direct sum.
is a balanced direct product.
is a direct product.
admits a non-trivial -action defined over .
admits a non-trivial -action defined over .
Moreover, if is a basis of in which factors as , then decomposes as
in the dual basis of .
Recall from [3, §3.1] that for a non-degenerate form , a decomposition
is called a maximally fine direct sum decomposition if , and is not a direct sum in , for all . For non-degenerate forms of degree , Kleppe has established that a maximally fine direct sum decomposition is unique [9, Theorem 3.7]. We use Theorem 1.6 to give an alternate proof of this result for smooth forms, deducing it from the fact that a polynomial ring over a field is a UFD:
Let . Suppose that either or . Let be a smooth form. Then has a unique maximally fine direct sum decomposition.
Let . Suppose is an algebraically closed field with . Then the following are equivalent for a GIT stable :
is a direct sum.
The morphism has a positive fiber dimension at .
has a -action.
is strictly semistable.
Consequently, the locus of direct sums is closed in the stable locus.
1.5. Prior works
In , Buczyńska, Buczyński, Kleppe, and Teitler prove that for a non-degenerate form over an algebraically closed field, the apolar ideal has a minimal generator in degree if and only if either is a direct sum, or is a limit of direct sums in which case the -orbit of contains an element of the form
where and are degree forms, in and variables, respectively. Since the form given by Equation (1.9) is visibly -unstable, and in particular singular, this translates into a computable and effective criterion for recognizing whether a smooth form is a direct sum over an algebraically closed field.
In , Kleppe uses the quadratic part of the apolar ideal to define an associative algebra of finite dimension over the base field ( is different from the Milnor algebra ). He then proves that, over an arbitrary field, direct sum decompositions of are in bijection with complete sets of orthogonal idempotents of .
A key step in the proof of the direct sum criterion in  is the Jordan normal form decomposition of a certain linear operator, which in general requires solving a characteristic equation. Similarly, finding a complete set of orthogonal idempotents requires solving a system of quadratic equations. This makes it challenging to turn  or  into an algorithm for finding direct sum decompositions when they exist.
The case of a linear factor in Theorem 3.1 was proved in [3, Proposition 2.12] using a criterion of Smith and Stong  for indecomposability of Gorenstein Artin algebras into connected sums. Our proof of the linear factor case and the statement for higher degree factors appear to be new. Corollaries 3.6 and 3.7 are generalizations of [3, Corollary 1.2], whose proof relies on a theorem of Shafiei  saying that the apolar ideals of the generic determinant and permanent are generated in degree ; our approach is independent of Shafiei’s results.
2. Proofs of decomposability criteria
Some implications in the statements of Theorems 1.6 and 1.8 are easy observations, the main of which is separated in Lemma 2.1 below. Others are found in recent papers [6, 7]. The remaining key ingredient that completes the main circle of implications is separated into Proposition 2.2 below.
Lemma 2.1 (No restrictions on ).
Suppose is a direct sum such that . Then the following hold:
There is a non-trivial one-parameter subgroup of that fixes and such that we have the following -weight-space decompositions
If , then is a balanced direct sum.
If , then there is a family of pairwise non-proportional forms in such that for all , and for all and .
This is obvious. Namely, suppose in some basis of , where . Then the one-parameter subgroup acting with weight on and weight on clearly satisfies (1).
Suppose further that . From
we see that and . Thus is a balanced direct sum. This proves (2). Taking for proves (3).
Proof of Theorem 1.6.
The implications are in Lemma 2.1. Next we prove . Suppose decomposes as a balanced direct sum in a basis of . Then
for some and .
It follows that for every and , we have
Using the assumption on , we conclude that
and so is a direct sum, in the same basis as .
The equivalence is proved in Proposition 2.2 below. This concludes the proof of equivalence for the first three conditions.
Next we prove . Suppose is a direct product decomposition in a basis of . Let be the dual basis of . Suppose is the smallest with respect to the graded reverse lexicographic order monomial of degree that does not lie in . Since must appear with a nonzero coefficient in , we have that . On the other hand, by [7, Lemma 4.1], we have that . It follows that . By symmetry, we also have that . We conclude that both inequalities must be equalities and so is a balanced direct product decomposition. Alternatively, we can consider a diagonal action of on that acts on as follows:
Then is homogeneous with respect to this action, and has weight However, the relevant parts of the proof of [6, Theorem 1.2] go through to show that satisfies the Hilbert-Mumford numerical criterion for semistability. This forces
We now turn to the last two conditions. First, the morphism is an -equivariant locally closed immersion by [2, §2.5], and so is stabilizer preserving. This proves the equivalence . The implication follows from the proof of [6, Theorem 1.0.1] that shows that for a smooth , the gradient point has a non-trivial -action if and only if is a direct sum. We note that even though stated over , the relevant parts of the proof of [6, Theorem 1.0.1] use only [6, Lemma 3.5], which remains valid over a field with or , and the fact that a smooth form over any field must satisfy the Hilbert-Mumford numerical criterion for stability. ∎
Proof of Theorem 1.8.
By [6, Theorem 1.0.1], for every GIT stable , the gradient point is polystable. Furthermore, it admits a -action if and only if is a direct sum. Moreover, [6, Proposition 2.1] shows that the morphism of the GIT quotients
is injective. This proves that for every stable , the fiber dimension of at equals to the dimension of the stabilizer of . This concludes the proof of all equivalences. The fact that the locus of direct sums is closed in now follows from the upper semicontinuity (on the domain) of fiber dimensions. ∎
Let . Suppose is a field with or . Then an element is a balanced direct sum if and only if is a balanced direct product. Moreover, if is a basis of in which factors as a balanced direct product, then decomposes as a balanced direct sum in the dual basis of .
The forward implication is an easy observation. Consider a balanced direct sum where and . Then, up to a nonzero scalar, , where and ; see [7, Lemma 2.11], which also follows from the fact that on the level of algebras, we have
Suppose now is a balanced direct product in a basis of :
where and . Let be the dual basis of , and let be the complete intersection ideal spanned by the elements of . We have that
It is then evident from (2.3) and the definition of an apolar ideal that
We also have the following observation:
By symmetry, it suffices to prove the second statement. Since is spanned by a length regular sequence of degree forms, we have that . Suppose we have a strict inequality. Let
Then is generated in degree , and has at least minimal generators in that degree. It follows that the top degree of is strictly less than , and so (cf. [7, Lemma 2.7]). But then
Using (2.5), this gives
Thus every monomial of
appears with coefficient in , which contradicts (2.3). ∎
At this point, we can apply [7, Prop. 3.1] to conclude that contains a regular sequence of length and that contains a regular sequence of length . This shows that decomposes as a balanced direct sum in the basis of . However, for the sake of self-containedness, we proceed to give a more direct argument:
By Claim (2.6), there exists a regular sequence such that
and a regular sequence such that
and let be the ideal generated by . We are going to prove that , which will conclude the proof of the proposition.
Since or , Macaulay’s theorem applies, and so to prove that , we need to show that the ideals and coincide in degree . For this, it suffices to prove that .
Since is a regular sequence in , we have that
Similarly, we have that
Set . It remains to show that
To this end, consider
where . Since , and we are working modulo , we can assume that , for all . Similarly, we can assume that , for all .
By construction, we have and . Using this, and (2.7), we conclude that
This finishes the proof of the proposition. ∎
Proof of Proposition 1.7.
If , the case of is vacuous since no smooth binary cubic will be a direct sum. In all other cases, implies .
Suppose are two maximally fine direct sum decompositions. Then
where . Suppose some shares irreducible factors with more than one . Then by the uniqueness of factorization in , we must have a non-trivial factorization such that . Then is a direct product, and so must be a direct sum by Theorem 1.6, contradicting the maximality assumption. Therefore, no shares an irreducible factor with more than one ; and, by symmetry, no shares an irreducible factor with more than one . It follows that and, up to reordering, , and thus , for all . We conclude that , which using forces , for all . ∎
3. Necessary conditions for direct sum decomposability
Our next two results give easily verifiable necessary conditions for an arbitrary form to be a direct sum. They hold over an arbitrary field, with no restriction on characteristic. We keep notation of §1.1.
Suppose is a form in .
Let . If has a factor such that , then is not a direct sum.
If has a repeated factor, then is not a direct sum.
Suppose is a form with . If has a linear factor, then is not a direct sum.
Proof of Theorem 3.1.
and that while . Let be the 1-PS subgroup of acting with weight on and weight on . Then fixes , and is the decomposition into the -weight-spaces. Since , we have
It follows by dimension considerations that some nonzero multiple of belongs to one of the two weight-spaces of in . Thus itself is homogeneous with respect to . It follows that either or . This forces either or , respectively. A contradiction!
For (2), suppose is a direct sum with a repeated factor . Let be the 1-PS of as above. Since , some nonzero multiple of belongs to a weight-space of in , and so we obtain a contradiction as in (1). ∎
Our next result needs the following:
Given a basis of and a nonzero , we define the state of to be the set of multi-indices such that
In other words, the state of is the set of monomials appearing with nonzero coefficient in . We set .
Suppose . Let . Suppose is such that in some basis of the following conditions hold: