PRIMARY DECOMPOSITION, HYPERDETERMINANTS AND JACOBIAN IDEALS OF TRILINEAR FORMS

Anna Guerrieri

(joint work with Irena Swanson)

Typeset by AMS-TEX

Primary decomposition

“… the earliest impulse toward the development of what is now commutative algebra, came from the desire of number theorists to make use of unique factorization in rings of integers in number fields other than ℚ. When it became clear that unique factorization did not always hold, the search for the strongest available alternative began. The theory of primary decomposition is the direct result of that search.”

D. Eisenbud - Commutative Algebra with a view toward Algebraic Geometry.

Among the names that built the theory of primary decomposition, one recalls:

Dedekind.

Lasker.

E. Noether: Hers is the highly nonconstructive intuition that the existence of primary decomposition depends only on the ascending chain condition for ideals. This approach simplified and clarified all the subject opening the way to the modern comprehension of the wide class of rings that now we call Noetherian in her honor.

It pushed, however, a little aside the taste (traceable in the work of Lasker) for effective, explicit results on primary decomposition.

Nowadays, it is instead this ‘computational flavor’ that it is at the center of most of the current investigations.

Let R be a Noetherian ring (e.g. a polynomial ring over a field) and let J be an ideal in R. The theory identifies a finite set of “associated” prime ideals of R, and tells how to “decompose” J as an intersection of “primary” ideals that are closely connected with these prime ideals.

A primary decomposition of J is:

J = Q_{1} ∩ ⋯ ∩ Q_{s},

where Q_{i} is a primary ideal (i.e. Q_{i} is a proper ideal and every zero-divisor in R/Q_{i} is nilpotent).

For each i there is a unique smallest prime ideal P_{i} containing Q_{i}. These primes are said to be the associated primes of J.

Eliminating the superfluous terms from the intersection and grouping together the Q_{k}’s corresponding to the same prime P_{i}, one makes the primary decomposition irredundant.

Examples:

• Let R = ℤ and J = (31752), then a primary decomposition of J is

J = (2^{3}) ∩ (3^{4}) ∩ (7^{2})

where the unique prime factorization of the integer 31752 is 2^{3} · 3^{4} · 7^{2}.

In this case the associated primes of J are the primes (2), (3) and (7). The primary components of J are (2^{3}), (3^{4}) and (7^{2}).

• Geometrically, if J is an ideal in a polynomial ring over an algebraically closed field (e.g. ℂ), primary decomposition theory produces a finite set of irreducible algebraic sets and says that J can be specified as the set of polynomial functions satisfying certain “vanishing conditions” on these irreducible parts.

In conlusion, attached to an ideal J in a Noetherian ring we have two lists:

• {P_{1}, …, P_{s}} - Associated Primes of J;

• {Q_{1}, …, Q_{s}} - Primary components of J.

Knowing the primary components one knows the associated primes too. It is possible, however, to know the associated primes without fully knowing the primary components.

These lists (perhaps the first is the most important) encrypt a great wealth of information about J.

What kind of information is encoded in the primary decomposition?

• Let I ⊆ J be ideals. Then I = J if and only if for every P associated prime of I, I_{P} = J_{P}.

• The zero-divisors of R/J are given by the union of all the associated primes of J.

Thus if one knows the associated primes of J, one also gains insight on the homological (geometrical) properties of R/J.

• Let (R, m) be an RLR (e.g. the ring of the formal power series in n variables over a field) and J an ideal of R, then the projective dimension of R/J is equal to the dimension of R if and only if m is an associated prime of J.

Recently, most of the research on primary decomposition focused on finding effective information. Without the ambition to be exhaustive, I summarize the recent efforts as follows:

Direct algorithms to compute the primary decomposition of an ideal:

Gianni, Träger and Zacharias (SINGULAR); Eisenbud, Huneke and Vasconcelos (MACAULAY

and SINGULAR); Shimoyama and Yokoyama (SINGULAR).

“The interplay between computation and many areas of algebra is a natural phenomenon in view of the algorithmic character of the latter. The existence of inexpensive but powerful computational resources has enhanced these links.”

W.V. Vasconcelos - Computational methods in commutative algebra and algebraic geometry Asymptotic behavior of primary components and associated primes:

Katz, Katzmann, Heinzer, Huneke, McAdam, Ratliff, Sharp, Swanson, etc.

“… behind the obvious finiteness condition in Noetherian rings, i.e., that every ideal is finitely generated, there lie many deeper and hidden types of finiteness which come to light in terms of uniform behavior.”

C. Huneke - Uniform bounds in Noetherian rings

Explicit primary decomposition of particular classes of ideals:

Here the list is ample. I just quote few classes of ideals and few names.

Monomial ideals: Bayer, Galligo and Stillman - Heinzer, Ratliff and Shah - Sturmfels, Trung and Vogel.

Binomial ideals: Eisenbud and Sturmfels.

Determinantal ideals: De Concini, Eisenbud and Procesi - Hochster - Huneke - etc.

Gaussian ideals: Corso, Vasconcelos and Villareal.

Jacobian ideals of multilinear forms: Boffi, Bruns and Guerrieri - Guerrieri and Swanson.

Permanental ideals: Laubenbacher and Swanson.

Applications to fields other than Algebra or Geometry:

Diaconis, Eisenbud and Sturmfels.

It turns out that primary decompositions of binomial ideals can give useful descriptions of components of a graph arising in problems from combinatorics, statistics, and operations research.

Hyperdeterminants and Jacobian ideals “The notion of determinant slowly emerged dur-

ing a period of one hundred and twenty years, from the pioneering work of Leibniz to the celebrated 1812 memoir of Cauchy … Approximately at the same time, people started to look for a multidimensional generalization of the notion of determinant, an idea that had already intrigued Vandermonde … A few years ago, Gelfand, Kapranov and Zelevinsky arrived at … a definition … motivated by the theory of general discriminants … But amazingly enough, they realized that Cayley (1845) had already defined the very same concept, while studying certain invariants of multilinear forms. He had even suggested to give the new notion the very appropriate name of hyperdeterminant.”

G. Boffi - On a Certain complex related to the notion of hyperdeterminant Let X denote the image of the Segre embedding:

ℙ^{n−1} × ℙ^{m−1} × ℙ^{p−1} :⟶ ℙ^{nmp−1}.

The hyperdeterminant of format n × m × p is a homogenous polynomial function on ℂ^{n−1}⊗ℂ^{m−1}⊗ ℂ^{p−1} which is a defining equation of the projectively dual variety X^{V} ( provided X^{V} is an hypersurface). If X^{V} is not an hypersurface, one sets the hyperdeterminant to be equal to 1 and refers to this case as trivial. The hyperdeterminant is denoted by Det.

• If n ≥ m ≥ p, the hyperdeterminant of format n × m × p is non-trivial if and only if n ≤ m + p − 1.

• Let (a_{ijk})_{\begin{matrix}i=1,... ,n \\ j=1,... ,m \\ k=1,... ,p \end{matrix}} be a a 3-dimensional complex

matrix (or array) of format n × m × p. This can be seen as a point of ℂ^{n−1} ×ℂ^{m−1} ×ℂ^{p−1}, so Det(A) is a polynomial function of the entries of the matrix.

Geometrically X^{V} is built as the closure of the set of all hyperplanes tangent to X.

Analytically, the hyperplane {f = 0} belongs to X^{V} if and only if f vanishes at some point of X with all its first derivatives.

If we choose coordinate systems X_{1}, …, X_{n} for (ℂ^{n−1})^{∗}, Y_{1}, …, Y_{m} for (ℂ^{m−1})^{∗} and Z_{1}, …, Z_{p} for (ℂ^{p−1})^{∗}, then f ∈ ℂ^{n−1} ⊗ ℂ^{m−1} ⊗ ℂ^{p−1} is represented after restriction on X by a trilinear form

f(\underline{X}, \underline{Y} , \underline{Z}) = A = ∑_{{1≤i≤n \atop 1≤j≤m} ; 1≤k≤p} a_{ijk}X_{i}Y_{j}Z_{k}

Thus, Det(A) = 0 means that the system of m + n + p + 1 equations

f = f_{X_{i}} = f_{Y_{j}} = f_{Z_{k}} = 0

has a solution (\underline{a}, \underline{b}, \underline{c}) with \underline{a}, \underline{b} and \underline{c} non-zero vectors.

In general, no formula for Det(A) is known, similar to the usual expression

∑_{σ∈S_{m}}(−1)^{sign σ}a_{1σ(1)} ⋯ a_{mσ(m)}

for the determinant of m×m matrices. In fact we are still far from fully understanding the combinatorial properties of the monomials occurring in Det(A).

In order to overcome this problem, various homological approaches to hyperdeterminants have been developed.

Boffi and Weyman analyzed a particular Koszul complex whose homology is affected by the vanishing of Det(A).

The 0-th homology of this complex is determined by the Jacobian ideal of A. Knowing the homological properties of such an ideal could shed light on this Koszul complex and, in turn, on Det(A).

In conclusion, the theory of hyperdeterminants, starting from the definitions to arrive to the latest homological approaches, suggests that interesting information could be trapped in the ideal generated by all the first partial derivatives of a trilinear form, the Jacobian ideal of the form.

In our work, we analyze the information coded in the possible primary decompositions of the Jacobian ideal of trilinear forms whose coefficients satisfy a weak kind of genericity.

Whatever I said about hyperdeterminants can be phrased in multilinear terms. It is just for the sake of semplicity that I restricted to the trilinear case.

Trilinear forms in general position

(joint work with Irena Swanson)

Let R = [X_{1}, …, X_{n}, Y_{1}, …, Y_{m}, Z_{1}, …, Z_{p}] with K a field. Assume n ≥ m ≥ p. Let

A = ∑_{{1≤i≤n \atop 1≤j≤m} ; 1≤k≤p} a_{ijk}X_{i}Y_{j}Z_{k}

be a trilinear form in R and let J_{A} be the ideal of R generated by the partial derivatives of A.

Together with Boffi and Bruns, I studied the associated primes of J_{A} when the coefficients of A identified a particular class of tridimensional arrays with non-zero hyperdeterminant:

• n = m + p − 1, (boundary condition),

• a_{ijk} ≠ 0 if and only if i = j + k − 1, (diagonal condition).

We asked if it was possible to relax these assumptions.

In the present work, we answer positively to this question.

In fact we show that the structure described in the previous work holds in a larger context.

The critical idea behind our results is the new concept of general position which we abstract from the proofs of Boffi, Bruns and myself. Thanks to this notion we eliminate the diagonal condition and relax the condition n = m + p − 1 to n ≥ m + p − 1.

Remarkably, when the equality holds, the class of trilinear forms we analyze identifies that of the three-dimensional arrays for which the hyperdeterminant is non-zero.

However, under strict inequality, it has no relation at all with the hyperdeterminants.

We fully exploit the interplay of three matrices of linear forms obtained by taking opportune second derivatives of A.

• Let X be the p × m matrix (A_{Y_{j}Z_{i}}).

• Let Y be the p × n matrix (A_{X_{j}Z_{i}}).

• Let Z be the m × n matrix (A_{X_{j}Y_{i}}).

The entries of X are linear forms in K[X_{1}, …, X_{n}], those of Y are linear forms in K[Y_{1}, …, Y_{m}] and those of Z are linear forms in K[Z_{1}, …, Z_{p}].

Notation

• \underline{X}, \underline{Y} and \underline{Z} denote (X_{1}, …, X_{n}), (Y_{1}, …, Y_{m}) and (Z_{1}, …, Z_{p}). Depending on the context, these stand for either the ideals or the row vectors.

• A_{X} stands for either the ideal or the vector (A_{X_{1}}, …, A_{X_{n}}). A_{Y} and A_{Z} are defined similarly.

With this notation we can express the first partial derivatives of A as products of matrices:

A_{X} = \underline{Z}Y = \underline{Y} Z, A_{Y} = \underline{Z}X = \underline{X}Z^{t}, A_{Z} = \underline{X}Y^{t} = \underline{Y} X^{t}.

For any matrix M and any integer q ≥ 0, I_{q}(M) stands for the ideal generated by the q × q minors of M. Thus we have

A_{X} = I_{1}(\underline{Z}Y) = I_{1}(\underline{Y} Z), A_{Y} = I_{1}(\underline{Z}X) = I_{1}(\underline{X}Z^{t}), A_{Z} = I_{1}(\underline{X}Y^{t}) = I_{1}(\underline{Y} X^{t}),

and some particular ideal of minors enter in the game:

\underline{Z}I_{p}(X) ⊆ I_{1}(\underline{Z}X) = A_{Y}, \underline{Z}I_{p}(Y), \underline{Y} I_{m}(Z) ⊆ I_{1}(\underline{Z}Y) = I_{1}(\underline{Y} Z) = A_{X}.

Before proceeding further let me give you a simple example.

A = X_{1}Y_{1}Z_{1} + X_{2}Y_{1}Z_{2} + X_{2}Y_{2}Z_{1} + X_{3}Y_{2}Z_{2} Here

Y = [ \begin{matrix}Y_{1} & Y_{2} & 0 \\ 0 & Y_{1} & Y_{2} \end{matrix} ] ,

X = [ \begin{matrix}X_{1} & X_{2} \\ X_{2} & X_{3} \end{matrix} ] ,

Z = [ \begin{matrix}Z_{1} & Z_{2} & 0 \\ 0 & Z_{1} & Z_{2} \end{matrix} ] .

Also

\begin{aligned}A_{X_{1}} & = Y_{1}Z_{1} \\ A_{X_{2}} & = Y_{1}Z_{2} + Y_{2}Z_{1} \\ A_{X_{3}} & = Y_{2}Z_{2} \end{aligned}

\begin{aligned}A_{Y_{1}} & = X_{1}Z_{1} + X_{2}Z_{2} \\ A_{Y_{2}} & = X_{2}Z_{1} + X_{3}Z_{2} \end{aligned} \begin{aligned}A_{Z_{1}} & = X_{1}Y_{1} + X_{2}Y_{2} \\ A_{Z_{2}} & = X_{2}Y_{1} + X_{3}Y_{2} \end{aligned}

A trilinear form A is said to be in general position when the conditions below are satisfied.

• n ≥ m ≥ p and n ≥ m + p − 1,

• I_{p}(X) has height m − p + 1 (maximal possible),

• I_{p}(Y) has height m (maximal possible),

• T^{−1}A_{Y} is prime of height m where T = {Z_{l}^{n}}_{n≥0}, 1 ≤ l ≤ p,

• T_{Z}^{−1}(A_{X}) = T_{Z}^{−1}(\underline{Y}), with T_{Z} = K[Z_{1}, …, Z_{p}] \ {0}.

Questions:

• Which trilinear forms are in general position?

• Is this notion linked to other already known concepts?

• What is the structure of J_{A} in this generalized setting?

The notion to which the concept of general position is linked is that of 1-genericity.

Let W_{1}, …, W_{s} be indeterminates over a field K. Let M be a q ×r matrix whose entries are linear forms in K[W_{1}, …, W_{s}].

We say that M is 1-generic if for any invertible row operation on M, the entries of each row generate an ideal of height min{r, s}.

In Linear sections of determinantal varieties, Eisenbud* introduced the concept of 1-genericity when s ≥ q + r − 1. In that case his definition and ours agree.

The 1 × r matrix [W_{1} ⋯ W_{s}0 ⋯ 0], where s < r, is 1-generic in our sense but not in Eisenbud’s

*D. Eisenbud spent a week at NMSU in 1996 as a Distinguished Visiting professor. He gave talks about Gale transforms and topics related to the concepts of 1-genericity and adjoints in the Eisenbud - Popescu sense.

When a matrix is 1-generic its ideal of minors are quite large.

Assume that K is algebraically closed.

• If M is a 1-generic q × r matrix in s variables W_{1}, …, W_{s} and s ≥ q+r−1, q ≤ r, then the height of I_{q}(M) is r − q + 1.

• If W is a q × r matrix whose entries are linear forms in the variables W_{1}, …, W_{s} and s, q ≤ r, then W is 1-generic if and only if the height of I_{q}(W) is maximal possible, namely s. Also, W is 1-generic if and only if the radical of I_{q}(W) is (W_{1}, …, W_{s}).

Example:

The field K needs to be algebraically closed. Let K = ℚ and Y_{1}, Y_{2} variables. Let Y be the 2 × 3 matrix

Y = ( \begin{matrix}Y_{1} & Y_{2} & 0 \\ Y_{2} & Y_{1} + Y_{2} & 0 \end{matrix} ) .

Each of the two rows generates (Y_{1}, Y_{2}), and for every b ∈ ℚ, the entries of (row 1) + b(row 2) generate

(Y_{1} +bY_{2}, Y_{2} +bY_{1} +bY_{2}) = (Y_{1} +bY_{2}, Y_{2}(1−b^{2} +b)).

As there is no rational number b for which 1−b^{2}+b = 0, this last ideal also has height 2 and Y is 1-generic. However, I_{2}(Y) is principal, so it cannot have height 2.

The following result highlights:

• the link between the notion of general position and that of 1-genericity,

• the interplay of X, Y and Z.

Theorem. Let n ≥ m + p − 1, n ≥ m ≥ p, and K an algebraically closed field. The following are equivalent:

(1) X is a 1-generic matrix.

(2) The transpose of X is a 1-generic matrix.

(3) Y is a 1-generic matrix.

(4) I_{p}(Y) has height m.

(5) \sqrt{I_{p}(Y)} = (Y_{1}, …, Y_{m}).

(6) Z is a 1-generic matrix.

(7) I_{m}(Z) has height p.

(8) \sqrt{I_{m}(Z)} = (Z_{1}, …, Z_{p}).

(9) A is a trilinear form in general position.

It is now possible to answer to the question: Which trilinear forms are in general position?

If K is an algebraically closed field:

• There exists a non-empty Zariski-open subset U of K^{nmp} such that if (a_{ijk}) ∈ U, then the corresponding A is in general position.

• When n = m + p − 1, A is in general position if and only if the hyperdeterminant of A is different from 0.

• The examples studied by Boffi, Bruns and myself are in general position.

Eventually, we may describe the structure of J_{A} in this wider setting.

Assume A is a trilinear form in general position:

The minimal primes of J_{A} (minimal among the associated primes) are:

• (Z_{1}, …, Z_{p}, A_{Z}), (Y_{1}, …, Y_{m}, A_{Y}) + I_{p}(X) when m > p;

• (Z_{1}, …, Z_{p}, A_{Z})+I_{p}(X), (Y_{1}, …, Y_{m}, A_{Y})+I_{p}(X), and (Y_{1}, …, Y_{m}, Z_{1}, …, Z_{p}) when m = p.

The radical of J_{A} is:

\sqrt{J_{A}} = J_{A} + \underline{Y} \underline{Z}.

On the primary components of J_{A} we have the following information:

• The minimal components are given by the minimal primes.

• J_{A} = \sqrt{J_{A}} ∩ (J_{A} + I_{p}(X) + I_{p}(Y) + I_{m}(Z)), and each embedded component of J_{A} contains J_{A} + I_{p}(X) + I_{p}(Y) + I_{m}(Z).

• If p = 2 and n = m + p − 1 = m + 1, the primary decomposition of J_{A} is

J_{A} = \sqrt{J_{A}} ∩ (\underline{X}, A_{X}, I_{2}(Y), I_{m}(Z)).

(We have an analogous result when p = 2 and n > m + p − 1.)

• We also compute the primary decomposition of various cases in which m = p = 3 and the matrix X is symmetric. The embedded primes all involve the ideals I_{t}(X) for t = 1, 2. The non-symmetric case has proved, so far, more elusive.