MAKSYM FEDORCHUK. n ) = z1 d 1 zn d 1.

Transcription

1 DIRECT SUM DECOMPOSABILITY OF SMOOTH POLYNOMIALS AND FACTORIZATION OF ASSOCIATED FORMS MAKSYM FEDORCHUK Abstract. We prove an if-and-only-if criterion for direct sum decomposability of a smooth homogeneous form in terms of the factorization properties of the Macaulay inverse system of its Milnor algebra. This criterion leads to an algorithm for computing direct sum decompositions over any field either of characteristic 0, or sufficiently large positive characteristic, for which polynomial factorization algorithms exist. 1. Introduction A homogeneous form F 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, e.g., F = F 1 (x 1,..., x a ) + F 2 (x a+1,..., x n ). A geometric significance of such a decomposition stems from the classical Thom-Sebastiani theorem [ST71] stating that if {F = 0} C n is an isolated singularity, then its monodromy operator is a tensor product of the monodromy operators of {F 1 = 0} C a and {F 2 = 0} C n a. In this paper, we give a new criterion for direct sum decomposability of smooth forms. The problem of finding such a criterion for an arbitrary (smooth or singular) form has been successfully addressed by Kleppe [Kle13] and Buczyńska-Buczyński-Kleppe-Teitler [BBKT15]. Both works interpret the direct sum decomposability of a form F in terms of its apolar ideal F (see 1.5 for more details). In particular, [BBKT15] gives an effective criterion for recognizing when F is a direct sum in terms of the graded Betti numbers of F. However, none of these works seem to give an effective algorithm for computing a direct sum decomposition when it exists. Our criterion accomplishes this for smooth forms. Recall that to a smooth homogeneous form F of degree d + 1 in n variables, one can assign a degree (d 1)n form A(F ) in n (dual) variables, called the associated form of F ([EI13, AI14, AI16]). The associated form A(F ) is defined as a Macaulay inverse system of the Milnor algebra of F [AI14], which simply means that the apolar ideal of A(F ) coincides with the Jacobian ideal of F : A(F ) = ( F/ x 1,..., 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 ) decomposes as a product of two forms in disjoint sets of (dual) variables. For example, up to a scalar, A(x d x d+1 n ) = z1 d 1 zn d 1. The purpose of this note is to prove the converse statement, and thus establish an ifand-only-if criterion for direct sum decomposability of a smooth form F in terms of the factorization properties of its associated form A(F ) (see Theorem 1.6). Although our criterion works only for smooth forms, it does so over an arbitrary field either of characteristic 0 or of sufficiently large characteristic, and it leads to an algorithm for finding direct sum decompositions over any such field for which polynomial factorization algorithms exist. This algorithm is given in Section 3. 1

2 2 MAKSYM FEDORCHUK 1.1. Notation and conventions. Let k be a field, and let V be a vector space over k with n := dim V 2. We set S := Sym V, and D := Sym V. We have a differentiation action of S on D. Namely, if x 1,..., x n is a basis of V, and z 1,..., z n is the dual basis of V, then the pairing S d D d D d d is given by G F = G( / z 1,..., / z n )F (z 1,..., z n ). If char(k) = 0 or char(k) > d, then the pairing S d D d k is perfect. Given a homogeneous non-zero F D d, the apolar ideal of F is F := {G S G F = 0} S. If char(k) = 0 or char(k) > d, the graded k-algebra S/F is a Gorenstein Artin local ring with socle in degree d. A well-known theorem of Macaulay establishes a bijection between graded Gorenstein Artin quotients S/I of socle degree d and elements of P(D d ) (see, e.g., [IK99, Lemma 2.12] or [Eis95, Exercise 21.7]). Remark 1.1. Even though we allow the field k to have positive characteristic, we do not take D to be the divided power algebra (cf. [IK99, Appendix A]), as the reader might have expected. The reason for this is that at several places we cannot avoid but to impose a condition that char(k) is large enough (or zero). In this case, the divided power algebra is isomorphic to D up to the needed degree. We say that a homogeneous form F S d+1 is smooth if the hypersurface V(F ) P n 1 is smooth over k. The locus of smooth forms in PS d+1 will be denoted by P(S d+1 ). The k-linear span of a subset W of a k-vector space will be denoted by W Direct sums and products. Recall from [BBKT15] that an element F Sym d+1 V is called a direct sum if there is a direct sum decomposition V = U W and non-zero F 1 Sym d+1 U and F 2 Sym d+1 W such that F = F 1 + F 2. In other words, F is a direct sum if and only if for some choice of a basis x 1,..., x n of V, we have that F = F 1 (x 1,..., x a ) + F 2 (x a+1,..., x n ), where 1 a n 1, and F 1, F 2 0. Recall also that a form F Sym d+1 V is called degenerate if there exists U V such that F Sym d+1 U. By analogy with direct sums, we will call a non-zero homogeneous form F D a direct product if there is a direct sum decomposition V = U W and F = F 1 F 2 for some F 1 Sym U and F 2 Sym W. In other words, a non-zero homogeneous F Sym D is a direct product if and only if for some choice of a basis z 1,..., z n of V, we have that (1.2) F (z 1,..., z n ) = F 1 (z 1,..., z a )F 2 (z a+1,..., z n ), where 1 a n 1. Furthermore, we call a direct product decomposition in (1.2) balanced if (n a) deg(f 1 ) = a deg(f 2 ). We say that [L] Grass(n, Sym d V ) is a balanced direct sum if there is a non-trivial direct sum decomposition V = U W and elements [L 1 ] Grass(dim U, Sym d U) and [L 2 ] Grass(dim W, Sym d W ) such that L = L 1 + L 2 Sym d U Sym d W Sym d V.

3 DECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS Associated forms. We briefly recall the theory of associated forms as developed in [AI16]. Let V act on S via the differentiation action. Then the gradient point of F S d+1 is defined to be F := F V S d, that is, the linear span of all first partial derivatives of F. If x 1,..., x n is a basis of V and F = F (x 1,..., x n ), then F = F/ x 1,..., F/ x n Sym d V. The Jacobian ideal of F is J F := ( F ) = ( F/ x 1,..., F/ x n ) S. Let Grass(n, S d ) Res be the affine open subset of complete intersections in Grass(n, S d ). Namely, Grass(n, S d ) Res parameterizes linear spaces g 1,..., g n S d such that g 1,..., g n form a regular sequence in S. Note that, if char(k) = 0 or char(k) > d + 1, then F S d+1 is smooth if and only if F Grass(n, S d ) Res. For every U = g 1,..., g n Grass(n, S d ) Res, the ideal I U = (g 1,..., g n ) is a complete intersection ideal, and the k-algebra S/I U is a graded Gorenstein Artin local ring with socle in degree n(d 1). Thus, by Macaulay s theorem, there exists a unique up to scaling form A(U) D n(d 1) such that (1.3) A(U) = I U. The form A(U) is called the associated form of g 1,..., g n by Alper and Isaev, who systematically studied it in [AI16, Section 2]. In particular, they showed 1 that the assignment U A(U) gives rise to an SL(n)-equivariant associated form morphism A: Grass(n, S d ) Res PD n(d 1). When U = F for a smooth form F S d+1, we set A(F ) := A( F ) and, following Eastwood-Isaev [EI13], call A(F ) the associated form of F. The defining property of A(F ) is that the apolar ideal of A(F ) is the Jacobian ideal of F : (1.4) A(F ) = J F S. This means that A(F ) is a Macaulay inverse system of the Milnor algebra M F := S/J F. Summarizing, when char(k) = 0 or char(k) > n(d 1), we have the following commutative diagram of SL(n)-equivariant morphisms: P ( S d+1 ) Grass ( n, S d )Res A P ( D n(d 1) ) A Remark 1.5. In [AI16], Alper and Isaev define the associated form A(g 1,..., g n ) as an element of D n(d 1), which they achieve by choosing a canonical generator of the socle of S/(g 1,..., g n ) given by the Jacobian determinant of g 1,..., g n. For our purposes, it will suffice to consider A( g 1,..., g n ) defined up to a scalar. 1 Although they work over C, their proof applies whenever char(k) = 0 or char(k) > n(d 1).

4 4 MAKSYM FEDORCHUK 1.4. Statement of main results. We have the following two results. Theorem 1.6. Let d 2. Suppose either char(k) = 0 or char(k) > n(d 1). Then the following are equivalent for a smooth F S d+1 : (1) F is a direct sum. (2) F is a balanced direct sum. (3) A(F ) is a balanced direct product. (4) F has a G m -action. (5) A(F ) has a G m -action. Moreover, if z 1,..., z n is a basis of V in which A(F ) factors as a balanced direct product A(F ) = G 1 (z 1,..., z a )G 2 (z a+1,..., z n ), then F decomposes as a direct sum F = F 1 (x 1,..., x a ) + F 2 (x a+1,..., x n ) in the dual basis x 1,..., x n of V. Theorem 1.7. Let d 2. Suppose k is a perfect field with either char(k) = 0 or char(k) > n(d 1). Then the following are equivalent for a GIT stable F S d+1 : (1) F is a direct sum. (2) F is strictly semistable. (3) dim ( SL(V ) F ) n 2 2. (4) A(F ) is strictly semistable. (5) dim ( SL(V ) A(F ) ) n Prior works. In [BBKT15], Buczyńska, Buczyński, Kleppe, and Teitler prove that for a non-degenerate form F k[x 1,..., x n ] d+1, the apolar ideal F has a minimal generator in degree d + 1 if and only if either F is a direct sum, or is a limit of direct sums in which case the GL(n)-orbit of F contains an element of the form (1.8) l i=1 x i H(x l+1,..., x 2l ) x l+i + G(x l+1,..., x n ), where H and G are degree d + 1 homogeneous forms, in l and n l variables, respectively. Since the form given by Equation (1.8) is visibly SL(n)-unstable, and in particular singular, this translates into a computable and effective criterion for recognizing whether a smooth form F is a direct sum. In [Kle13], Kleppe uses the quadratic part of the apolar ideal F to define an associative algebra M(F ) of finite dimension over k (which is different from the Milnor algebra M F defined above). He then proves that direct sum decompositions of F are in bijection with complete sets of orthogonal idempotents of M(F ). A key step in the proof of the direct sum criterion in [BBKT15] 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 [BBKT15] and [Kle13] into an algorithm for finding direct sum decompositions when they exist, in the case of non-finite fields.

5 DECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 5 2. Proofs Some implications in the statements of Theorems 1.6 and 1.7 are easy observations. Others are found in the recent papers [Fed17, FI17]. The remaining key ingredient that completes the main circle of implications is separated into Proposition 2.2 below. Proof of Theorem 1.6. The implications (1) = (2) = (4) are obvious. Next we prove (2) = (1). Suppose F decomposes as a balanced direct sum in a basis x 1,..., x n of V. Then F = (s 1,..., s a, t 1,..., t n a ), for some s 1,..., s a k[x 1,..., x a ] d and t 1,..., t n a k[x a+1,..., x n ] d. It follows that for every 1 i a and a + 1 j n, we have 2 F x i x j k[x 1,..., x a ] d 1 k[x a+1,..., x n ] d 1 = (0). Using the assumption on char(k), we conclude that F k[x 1,..., x a ] d+1 k[x a+1,..., x n ] d+1, and so is a direct sum (in the same basis as F ). The equivalence (2) (3) is proved in Proposition 2.2 below (the forward implication being quite straightforward). This concludes the proof of equivalence for the first three conditions. We now turn to the last two conditions. First, the morphism A is an SL(n)-equivariant locally closed immersion by [AI16, 2.5], and so is stabilizer preserving. This proves the equivalence (4) (5). The implication (4) = (1) follows from the proof of [Fed17, Theorem 1.0.1] that shows that for a smooth F, the gradient point F has a non-trivial G m -action if and only if F is a direct sum. We note that even though stated over C, the relevant parts of the proof of [Fed17, Theorem 1.0.1] use only [Fed17, Lemma 3.5] that remains valid over a field k with char(k) = 0 or char(k) > d+1 and the fact that a smooth form over any field must satisfy the Hilbert-Mumford numerical criterion for stability. Remark 2.1. Given a basis of V diagonalizing a non-trivial G m -action on F, the proof of [Fed17, Theorem 1.0.1] can be turned into an algorithm for finding a basis of V in which F decomposes into a direct sum. At the same time, using a tangent space computation, it is possible to give a purely linear-algebraic criterion for F to admit a non-trivial G m action, without writing one down explicitly. Proof of Theorem 1.7. Since the field k is taken to be perfect, the Hilbert-Mumford numerical criterion for GIT semistability (resp., polystability) holds. By [Fed17, Theorem 1.0.1], for every GIT stable F, the gradient point F is polystable and is striclty semistable if and only if F is decomposable. 2 This proves the equivalences (1) (2) (3). By [FI17, Theorem 4.1], the morphism A, and hence A, preserves polystability. Together with the fact that A is stabilizer preserving, this proves the equivalences (4) (5) (3). 2 Again, although [Fed17, Theorem 1.0.1] is stated over C, the proof goes through for any perfect field of characteristic either 0 or greater than d + 1.

6 6 MAKSYM FEDORCHUK Proposition 2.2. Let d 2. Suppose k is a field with char(k) = 0 or char(k) > n(d 1). Then an element U Grass(n, Sym d V ) Res is a balanced direct sum if and only if A(U) is a balanced direct product. Moreover, if z 1,..., z n is a basis of V in which A(U) factors as a balanced direct product, then U decomposes as a balanced direct sum in the dual basis x 1,..., x n of V. Proof. The forward implication is an easy observation. Indeed, consider a balanced direct sum U = g 1,..., g n Grass(n, k[x 1,..., x n ] d ) Res, where g 1,..., g a k[x 1,..., x a ] d and g a+1,..., g n k[x a+1,..., x n ] d, then A(U) = A(g 1,..., g a )A(g a+1,..., g n ), up to a non-zero scalar, by, e.g., [FI17, Lemma 2.16], which follows from the fact that on the level of algebras, we have k[x 1,..., x n ] (g 1,..., g n ) k[x 1,..., x a ] (g 1,..., g a ) k[x a+1,..., x n ] k (g a+1,..., g n ). Suppose now A(U) decomposes as a direct product in a basis z 1,..., z n of V : (2.3) A(U) = F 1 (z 1,..., z a )F 2 (z a+1,..., z n ), where deg(f 1 ) = a(d 1) and deg(f 2 ) = (n a)(d 1). Let x 1,..., x n be the dual basis of V, and let I U k[x 1,..., x n ] be the complete intersection ideal spanned by the elements of U. Then I U = A(U) k[x 1,..., x n ]. It is then evident from (2.3) and the definition of an apolar ideal that (2.4) (x 1,..., x a ) a(d 1)+1 I U and (2.5) (x a+1,..., x n ) (n a)(d 1)+1 I U. We also have the following observation: Claim 2.6. dim k ( U (x1,..., x a ) ) = a, dim k ( U (xa+1,..., x n ) ) = n a. Proof. By symmetry, it suffices to prove the second statement. Since U is spanned by a length n regular sequence of degree d forms, we have that dim k ( U (xa+1,..., x n ) ) n a. Suppose we have a strict inequality. Let R := k[x 1,..., x n ]/(I U, x a+1,..., x n ) k[x 1,..., x a ]/I. Then I is generated in degree d, and has at least a + 1 minimal generators in that degree. It follows that the top degree of R is strictly less than a(d 1), and so I a(d 1) = k[x 1,..., x a ] a(d 1) (cf. [FI17, Lemma 2.15]). But then k[x 1,..., x a ] a(d 1) (x a+1,..., x n ) mod I U.

7 Using (2.5), this gives Thus every monomial of DECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 7 k[x 1,..., x a ] a(d 1) k[x a+1,..., x n ] (n a)(d 1) I U. k[z 1,..., z a ] a(d 1) k[z a+1,..., z n ] (n a)(d 1) appears with coefficient 0 in A(U), which contradicts (2.3). By Claim (2.6), there exists a regular sequence s 1,..., s a k[x 1,..., x a ] d such that (s 1,..., s a ) = ( g 1 (x 1,..., x a, 0,..., 0),..., g n (x 1,..., x a, 0,..., 0) ) and a regular sequence t 1,..., t n a k[x a+1,..., x n ] d such that (t 1,..., t n a ) = ( g 1 (0,..., 0, x a+1,..., x n ),..., g n (0,..., 0, x a+1,..., x n ) ). Let W := s 1,..., s a, t 1,..., t n a Grass(n, S d ) Res and let I W be the ideal generated by W. We are going to prove that U = W, which will conclude the proof of the proposition. Since char(k) = 0 or char(k) > n(d 1), Macaulay s theorem applies, and so to prove that U = W, we need to show that the ideals I U and I W coincide in degree n(d 1). For this, it suffices to prove that (I W ) n(d 1) (I U ) n(d 1). Since s 1,..., s a is a regular sequence of elements in k[x 1,..., x n ] d, we have that Similarly, we have that Together with (2.4) and (2.5), this gives k[x 1,..., x a ] a(d 1)+1 (s 1,..., s a ). k[x a+1,..., x n ] (n a)(d 1)+1 (t 1,..., t n a ). (2.7) (x 1,..., x a ) a(d 1)+1 + (x a+1,..., x n ) (n a)(d 1)+1 I U I W. Set J := (x 1,..., x a ) a(d 1)+1 + (x a+1,..., x n ) (n a)(d 1)+1. It remains to show that To this end, consider (I W ) n(d 1) (I U ) n(d 1) mod J. a i=1 n a q i s i + r j t j (I W ) n(d 1), j=1 where q 1,..., q a, r 1,..., r n a k[x 1,..., x n ] n(d 1) d. Since s 1,..., s a k[x 1,..., x a ] d, and we are working modulo J, we can assume that q i (x a+1,..., x n ) (n a)(d 1), for all i = 1,..., a. Similarly, we can assume that r j (x 1,..., x a ) a(d 1). By construction, we have that s 1,..., s a I U mod (x a+1,..., x n ) and t 1,..., t n a I U mod (x 1,..., x a ). Using this, and (2.7), we conclude that a i=1 n a q i s i + r j t j I U mod J. j=1 This finishes the proof of the proposition.

8 8 MAKSYM FEDORCHUK 3. Finding a balanced direct product decomposition algorithmically In this section, we show how Theorem 1.6 reduces the problem of finding a direct sum 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 easily computed in the dual basis of V as the form apolar to (J F ) n(d 1). To apply Theorem 1.6, we now need to determine if A(F ) decomposes as a balanced direct product, and if it does, then in what basis of V. The following simple lemma explains how to do it. Lemma 3.1. An associated form A(F ) is a balanced direct product if and only if there is a non-trivial factorization A(F ) = G 1 G 2 such that (3.2) V = ( G ) 1 1 ( G ) 2 1. Moreover, A(F ) decomposes as a balanced direct product in any basis of V such that its dual basis is compatible with the direct sum decomposition in Equation (3.2). Proof. Indeed, suppose that for some choice of a basis z 1,..., z n of V, we have that G = G 1 (z 1,..., z a )G 2 (z a+1,..., z n ) is a balanced direct product. Let x 1,..., x n be the dual basis of V. Then clearly {x a+1,..., x n } ( G ) 1 and {x 1,..., x a } ( G ) 2 1. We also have that ( ) G 1 1 ( G ) 2 1 ( A(F ) ) 1 = (J F ) 1 = (0). This proves that V = ( G ) 1 1 ( G ) 2 1. Suppose now V = ( G ) 1 1 ( G ) 2 1 for some non-trivial factorization A(F ) = G 1G 2. Choose a basis x 1,..., x n of V such that ( G ) 1 1 = x a+1,..., x n and ( G ) 2 1 = x 1,..., x a. Let z 1,..., z n be the dual basis of V. Since char(k) = 0 or char(k) > deg(g 1 ), deg(g 2 ), we have that in the basis z 1,..., z n, It follows that G 1 = G 1 (z 1,..., z a ), G 2 = G 2 (z a+1,..., z n ). (3.3) G = G 1 (z 1,..., z a )G 2 (z a+1,..., z n ). It remains to show that the direct product decomposition in (3.3) is balanced. To this end, let x d 1 1 xdn n be the smallest with respect to the grevlex order monomial of degree n(d 1) that does not lie in (J F ) n(d 1). Since z d 1 1 zdn n must appear with a non-zero coefficient in A(F ), we have that d d a = deg G 1. On the other hand, by [FI17, Lemma 4.5], we have that d d a a(d 1). It follows that deg G 1 a(d 1). By symmetry, we also have that deg G 2 (n a)(d 1). We conclude that both inequalities must be equalities and so (3.3) is a balanced direct product decomposition of A(F ). 1

9 DECOMPOSABILITY OF POLYNOMIALS AND ASSOCIATED FORMS 9 Remark 3.4. Alternatively, we can consider a diagonal action of G m SL(V ) on V that acts on V as follows: ) t (z 1,..., z n ) = (t (n a) z 1,..., t (n a) z a, t a z a+1,..., t a z n. Then G is homogeneous with respect to this action, and has weight (n a) deg G 1 a deg G 2. However, the relevant parts of the proof of [Fed17, Theorem 1.2] go through to show that A(F ) satisfies the Hilbert-Mumford numerical criterion for semistability. This forces (n a) deg G 1 a deg G 2 = An algorithm for finding direct sum decompositions. Let F k[x 1,..., x n ] d+1 where d 2. Suppose k is a field, with either char(k) = 0 or char(k) > n(d 1), and over which there exists a polynomial factorization algorithm. Step 1: Compute J F = ( F/ x 1,..., F/ x n ) up to degree n(d 1) + 1. If (J F ) n(d 1)+1 k[x 1,..., x n ] n(d 1)+1, then F is not smooth and we stop; otherwise, continue. Step 2: Compute A(F ) as the dual to (J F ) n(d 1) : A(F ) = { T k[z 1,..., z n ] n(d 1) G( / z 1,..., / z n ) T = 0, for all G (J F ) n(d 1) }. Step 3: Factor A(F ) into irreducible components in k[z 1,..., z n ] and check for the existence of balanced direct product factorizations using Lemma 3.1. 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 3.1 gives a basis of V in which F decomposes as a direct sum. Remark 3.5. Jarek Buczyński has pointed out that Step 2 in the above algorithm is computationally highly expensive when n and d are large. References [AI14] Jarod Alper and Alexander Isaev, Associated forms in classical invariant theory and their applications to hypersurface singularities, Math. Ann. 360 (2014), no. 3-4, MR [AI16], Associated forms and hypersurface singularities: The binary case, J. reine angew. Math. (2016), To appear, DOI: /crelle [BBKT15] Weronika Buczyńska, Jaros law Buczyński, Johannes Kleppe, and Zach Teitler, Apolarity and direct sum decomposability of polynomials, Michigan Math. J. 64 (2015), no. 4, MR [EI13] Michael Eastwood and Alexander Isaev, Extracting invariants of isolated hypersurface singularities from their moduli algebras, Math. Ann. 356 (2013), no. 1, MR [Eis95] David Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, MR [Fed17] Maksym Fedorchuk, GIT semistability of Hilbert points of Milnor algebras, Math. Ann. 367 (2017), no. 1-2, MR [FI17] Maksym Fedorchuk and Alexander Isaev, Stability of associated forms, 2017, arxiv: [math.ag].

10 10 MAKSYM FEDORCHUK [IK99] Anthony Iarrobino and Vassil Kanev, Power sums, Gorenstein algebras, and determinantal loci, Lecture Notes in Mathematics, vol. 1721, Springer-Verlag, Berlin, MR [Kle13] J. Kleppe, Additive Splittings of Homogeneous Polynomials, 2013, arxiv: [math.ag]. [ST71] M. Sebastiani and R. Thom, Un résultat sur la monodromie, Invent. Math. 13 (1971), MR (Fedorchuk) Department of Mathematics, Boston College, 140 Commonwealth Ave, Chestnut Hill, MA 02467, USA address: