Direct sum decomposability of polynomials and factorization of associated forms

Abstract

We prove two criteria for direct sum decomposability of homogeneous polynomials. 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 0 or of sufficiently large positive characteristic, for which polynomial factorization algorithms exist. For homogeneous forms over algebraically closed fields, we interpret direct sums and their limits as forms that cannot be reconstructed from their Jacobian ideal. 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.

Deformation of Milnor algebras

“…By [6,Proposition 4.8 or Corollary 3.15], a smooth homogeneous polynomial f ∈ S d admits a unique maximally fine "direct sum decomposition" (7) f (x 0 , · · · , x n ) = f 1 (x 0 , · · · , x n 1 −1 ) + f 2 (x n 1 , · · · , x n 2 ) + · · · + f s (x n s−1 , · · · , x n ), for a choice of linear coordinates {x i } n i=0 , where 0 ≤ n 1 ≤ n 2 ≤ · · · ≤ n s−1 ≤ n and none of the f j 's is of ST type. In addition, if g ∈ S d satisfies E d−1 (g) ⊂ E d−1 (f ), then necessarily, g is of the following form (8) g = λ 1 f 1 + · · · + λ s f s , λ i ∈ C, see [6,Corollary 3.12]. In particular, if g is also smooth, then all the λ j 's in (8) are nonzero.…”

“…We say that a polynomial f ∈ S d is of Sabastiani-Thom type (ST type) or a direct sum if f can be represented as (1) f (x 0 , · · · , x n ) = f 1 (x 0 , · · · , x ℓ ) + f 2 (x ℓ+1 , · · · , x n ) for a choice of homogeneous coordinates {x i } n i=0 of P n and some 0 ≤ ℓ < n; see [13,14,2]. For various characterizations of polynomials of ST type, we refer to [6]. Denote by U ⊂ P(S d ) ∆ the set of all smooth homogeneous polynomials that are not of ST type.…”

Associated Form Morphism

“…Let DS ss d+1 := P(DS d+1 ) ss be the locus of semistable direct sums in P S d+1 ) ss . By [8,Section 3], the set DS ss d+1 is precisely the closed locus in P S d+1 ) ss where ∇ has positive fiber dimension.…”

“…We will use the recognition criteria for direct sums established in [8], and so we keep the pertinent terminology of that paper. We will say that f ∈ S m is a k-partial Fermat form for some k ≤ n, if, after a linear change of variables, it can be written as follows: f = x m 1 + · · · + x m k + g(x k+1 , .…”

On Centers and Direct Sum Decompositions of Higher Degree Forms

“…On the other hand, recently direct sum decompositions of higher degree forms are also approached through apolarity, see for example [8,1,17]. In [2], the direct sum decomposition of a smooth form is interpreted in terms of the product factorization of its associated form and an algorithm for computing direct sum decompositions is provided. In these aforementioned works, criteria and algorithms of direct sum decompositions of higher degree forms involve sophisticated tools of Gröbner bases and associated forms.…”