The monic rank

The monic rank

Arthur Bik Universität Bern, Mathematisches Institut, Alpeneggstrasse 22, 3012 Bern arthur.bik@math.unibe.ch Jan Draisma Universität Bern, Mathematisches Institut, Sidlerstrasse 5, 3012 Bern, and Eindhoven University of Technology jan.draisma@math.unibe.ch Alessandro Oneto Barcelona Graduate School of Mathematics, and Universitat Politècnica de Catalunya, Av.da Diagonal 647 (ETSEIB), 08028 Barcelona, Spain alessandro.oneto@upc.edu, aless.oneto@gmail.com  and  Emanuele Ventura Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA eventura@math.tamu.edu, emanueleventura.sw@gmail.com
Abstract.

We introduce the monic rank of a vector relative to an affine-hyperplane section of an irreducible Zariski-closed affine cone . We show that the monic rank is finite and greater than or equal to the usual -rank. We describe an algorithmic technique based on classical invariant theory to determine, in concrete situations, the maximal monic rank. Using this technique, we establish three new instances of a conjecture due to Shapiro which states that a binary form of degree is the sum of -th powers of forms of degree . Furthermore, in the case where is the cone of highest weight vectors in an irreducible representation—this includes the well-known cases of tensor rank and symmetric rank—we raise the question whether the maximal rank equals the maximal monic rank. We answer this question affirmatively in several instances.

AB was supported by JD’s Vici grant. JD was partially supported by the NWO Vici grant entitled Stabilisation in Algebra and Geometry. AO acknowledges financial support from the Spanish Ministry of Economy and Competitiveness, through the María de Maeztu Programme for Units of Excellence in RD (MDM-2014-0445). EV acknowledges financial support by the grant 346300 for IMPAN from the Simons Foundation and the matching 2015-2019 Polish MNiSW fund. AO and EV thank the Universität Bern (Switzerland) for the hospitality during visits where the project of the paper was discussed.

1. Introduction

Let be an algebraically closed field of characteristic zero. All our vector spaces and algebraic varieties will be over , finite-dimensional, reduced and identified with their sets of -points.

Monic secant varieties and monic rank

Let be a finite-dimensional vector space. Let be an irreducible Zariski-closed affine cone such that its -linear span equals , i.e., the cone is non-degenerate. A very fruitful field of research investigates the concept of -rank of an element , denoted by , i.e., the minimal number of elements of such that lies on their linear span (see [BT15] for definitions). In the cases where is, for example, some vector space of tensors and is the subvariety of rank tensors in , the study of -ranks has very interesting relations with fields in applied mathematics. We refer to [Lan12] for a more extensive exposition. In this paper we introduce a new, but related, type of rank.

Let be a non-zero linear function and consider the affine hyperplane

We write for the affine-hyperplane section of .

Definition 1.1.

Let be a positive integer. The -th open secant variety of is the set

This is a subset of . We define to be the Zariski closure of and call this set the -th secant variety of . We also call the -th open monic secant variety of and the -th monic secant variety of .

Since is an affine space, we have . And, for any , we have

However as we shall see, both inclusions can be strict. We now define the monic rank of a vector with .

Definition 1.2.

Let be a vector. The monic rank of is defined to be

Similarly, the monic border rank of is

The following example is illustrative for the rest of our results.

Example 1.3.

Let be the vector space of binary forms of degree and let be the subset of squares of linear forms. We consider the linear function which selects the coefficient of . So

We get and . Now, an element of is contained in the second open monic secant if and only if it equals

for some . Notice that the polynomials and generate the ring of symmetric polynomials in the variables and , i.e., the invariant ring . Here is the symmetric group on two letters and acts by permuting and . From classical invariant theory, we know that the map

is a finite morphism. Thus, it is closed and dominant and so it is also surjective. Hence

See Proposition 3.3 and the proof of Theorem 1.6 for an explanation. We find that any satisfies .

Main results

A priori it is not clear that the monic rank of an element of is even finite. This is our first foundational result.

Theorem 1.4.

The function is strictly increasing until it coincides with its maximal value and constant from then on. Let be the minimal integer for which holds. Then for any , we have

In particular, the monic rank is finite.

Definition 1.5.

The minimal integer for which is called the generic monic rank of elements of .

Theorem 1.4 mimics a result of Blekherman and Teitler [BT15, Theorem 3] that relates the maximal -rank to the generic -rank.

An interesting conjecture due to Shapiro (see [LORS18, Conjecture 1.4]) states that every binary form of degree is the sum of -th powers of forms of degree . We show that, in a few particular cases, a stronger statement holds: every binary form of degree whose coefficient at equals is the sum of -th powers of monic forms of degree .

Theorem 1.6.

Let and be positive integers. Let be the space of binary forms of degree , take and let be the linear function that maps a form to its coefficient at . Then, in the following cases, the maximal monic rank is at most :

  1. and arbitrary ;

  2. and arbitrary ;

  3. and ;

  4. and .

In particular, in these cases, the -rank of any is at most .

In terms of the (non-monic) Shapiro’s conjecture on writing binary forms of degree as sums of -th powers:

  • the case is classical (see, e.g., [Rez13, Theorem 4.9]);

  • the case is trivial;

  • the case is quite immediate (see [FOS12, Theorem 5]); and

  • the case was proven in [LORS18, Theorem 3.1].

As far as we know, the cases where are new.

We next turn our attention to a particularly nice class of varieties. Let be a connected reductive algebraic group over and let be an irreducible finite-dimensional rational representation of . Let be the cone of highest-weight vectors, i.e., the affine cone over the unique closed -orbit in . The latter projective variety is called a homogeneous variety. Let be a highest weight vector in the dual -module .

Question 1.7.

In the setting above, is the maximal -rank of a vector in equal to the maximal monic -rank of a vector in ?

Let be a finite-dimensional vector space. Then we denote the -th symmetric power of  by .

Theorem 1.8.

The answer to Question 1.7 is affirmative in the following instances:

  1. and ;

  2. and ;

  3. and ;

  4. and ; and

  5. and is its adjoint representation.

Remark 1.9.

The first case of the theorem corresponds to the Waring rank of binary forms, which is also (i) in Theorem 1.6. The second case corresponds to the usual matrix rank of matrices. Case (iii) corresponds to the symmetric matrix rank of symmetric matrices (or to the Waring rank of quadrics in variables). Case (iv) corresponds to the tensor rank of tensors. And lastly, case (v) corresponds to the -rank of trace-zero matrices with being the affine cone over the projective adjoint variety of incident point-hyperplane pairs in .

Admittedly, this is not much evidence for an affirmative answer to Question 1.7 in general. Moreover, our proofs in each of these cases are ad hoc. We therefore appeal to the reader for different approaches to Question 1.7. In particular, an affirmative answer to that question for and would imply that a new lower bound of

on the maximal rank of a form of degree . This is almost always bigger than

which (except for quadrics and finitely many further exceptions) is the generic Waring rank by the Alexander-Hirschowitz theorem [AH95]. So then the maximal rank would be greater than the generic rank—a commonly held belief to which no approach is known.

Structure of the paper

In Section 2, we lay the foundations of the notion of monic rank and prove Theorem 1.4. In Section 3, we develop some machinery from classical invariant theory to compute the maximal monic rank in certain explicit cases. We apply this machinery to the proof of Theorem 1.6 in Section 4. Finally, in Section 5, we establish Theorem 1.8.

2. The basics of monic rank

Recall the following notation from the introduction:

  • denotes a non-degenerate irreducible affine cone;

  • is a non-zero linear function; and

  • is an affine hyperplane and .

Theorem 1.4 will be a direct consequence of Proposition 2.1 and Proposition 2.2.

Proposition 2.1.

The function is strictly increasing until it coincides with its maximal value, which is , and constant from then on. Consequently, the function is bounded. Moreover, its value at is an upper bound on the ordinary border -rank of for all vectors .

Proof.

Let be any point. Then . So the function is weakly increasing. Since is bounded from above by , there exists a such that . Let be any positive integer with this property. Then, since both and are irreducible, the isomorphism

restricts to an isomorphism between and . By definition, a general point on is of the form with and and we have

Therefore, the isomorphism

restricts to an isomorphism between and . Now, let be the minimal value of for which . Then we conclude that the function is strictly increasing for and is constant for .

Next, we show that , which implies in particular that . Let be positive integers. Then

Since the leftmost and rightmost sets are closed, irreducible and of the same dimension, all three sets coincide. For any , we have . Therefore

So the line through and intersects in infinitely many points. Hence this line must be entirely contained in . Since this holds for all , we see that is an affine space. Since is non-degenerate, the affine span of coincides with . So the affine span of equals . We conclude that .

For the last statement, note that is contained in the ordinary -th secant variety of and that the ordinary border rank of a vector does not change whenever we multiply the vector by a non-zero constant. ∎

Proposition 2.2.

Let be the generic monic rank of elements of . Then, for every vector , we have . In particular, the monic rank of is finite.

Proof.

We adopt a similar strategy as in the proof of [BT15, Theorem 3]. Recall that the generic monic rank is finite by Proposition 2.1. Set

and consider the intersection

Both sets on the left-hand side contain an open dense subset of . Thus they must intersect. Consequently, there exist such that . So

and hence . This shows the second inequality. The first is immediate from the definitions of -rank and monic -rank. ∎

3. Invariant theory tools

Our reference for classical invariant theory is [DK02].

3.1. A variant of a theorem of Hilbert

In this section, we develop computational tools to prove Theorem 1.6. Let be a reductive algebraic group over acting on an affine variety , whose coordinate ring is . Let a one-dimensional torus act on . The character lattice of is isomorphic to . For any , let be the corresponding weight space:

This naturally induces a grading on . Assume the following:

  1. The grading satisfies and for .

  2. The actions of and on commute.

Under these assumptions, each weight space is a representation of and the invariant ring decomposes as

where . In this section, the terms homogeneous and degree refer exclusively to the grading given by .

Proposition 3.1.

Suppose that are homogeneous elements of positive degree such that

where denotes the vanishing set of a set of forms . Then is a finitely generated module over its subring .

Remark 3.2.

In Hilbert’s classical variant of this result (see, e.g., [DK02, Lemma 2.4.5]), the variety is a vector space, acts linearly on and the grading is the standard one. The proof of our generalization is identical, but we include it for the sake of completeness.

Proof.

Let be homogeneous generators of of positive degree. Then by Hilbert’s Nullstellensatz, there exists a positive integer such that, for each , the form is in the ideal generated by the homogeneous forms . Using the Reynolds operator, one finds that

for some homogeneous with . So we see that the finite set

generates as a -submodule of . ∎

3.2. A criterion for closedness of the open monic secant variety

Recall that we have a non-degenerate irreducible affine cone and a linear function , the affine hyperplane and the affine-hyperplane section . Assume a one-dimensional torus acts linearly on such that the following conditions are satisfied:

  • The action extends to a morphism (or equivalently, all weights of are nonnegative).

  • The space of -invariant linear forms is spanned by .

  • The set is stable under .

Let be a basis of consisting of -weight vectors (i.e., a basis of homogeneous vectors). Since is -invariant, the sets and are stable under . The affine section contains the unique -fixed point . The coordinates of in the given basis of are .

We now fix a positive integer , let be the -fold direct product over of the affine variety with itself and we consider the following map:

Besides the induced action of , the affine variety comes naturally equipped with an action of the symmetric group which permutes the factors. Note that the actions of and on commute by definition and that is -equivariant and -invariant.

Proposition 3.3.

In the setting above, if , then is closed and of dimension .

Proof.

Let be the coordinate ring of . Define to be the affine variety whose coordinate ring is the invariant ring , i.e., . Since is -invariant, it factors through the quotient map . So we have a morphism that makes the diagram

span

commutative. Note that .

A classical result [DK02, Lemma 2.3.2] in invariant theory is that the quotient map  is surjective. We show that the vertical map is closed. Since is -equivariant, the map is -equivariant as well. Let be the pull-back map induced by  and take for each . By definition, we have

for all for every . Here we use that the pull-back map induced by is the inclusion . We see that

This means that all these subsets of coincide. So by Proposition 3.1, the morphism is finite and hence closed. This implies that the image of is closed. It also implies that the dimension of coincides with the one of , which is . ∎

4. Instances of Shapiro’s conjecture

Fix positive integers and . As a first application of the monic rank, we look at a conjecture due to Shapiro.

Conjecture 1 (Shapiro’s Conjecture, [Lors18, Conjecture 1.4]).

Every binary form of degree can be written as the sum of -th powers of forms of degree .

We prove this conjecture in some new cases by proving the following (stronger) conjecture: let be the vector space of binary forms of degree , let

be the variety of -th powers of forms of degree , let be the affine hyperplane consisting of all forms whose coefficient at equals and take .

Conjecture 2.

The addition map

satisfies .

Proposition 4.1.

If Conjecture 2 holds for , then is closed and of dimension . In particular, Conjecture 2 for implies Shapiro’s Conjecture for .

Proof.

Let act on via

for all . This action has only positive weights and stabilizes . Furthermore, the only -invariant in is the linear function which selects the coefficient of . Hence, the assumptions of Proposition 3.3 are satisfied. This implies the first statement. For the second statement, note that every non-zero form has in its -orbit a form with . Assuming Conjecture 2, we derive

as desired. ∎

Remark 4.2.

Let be a triple such that Conjecture 2 holds for and let be monic binary forms with . Then we have

and hence by the conjecture for . So we see that the conjecture for implies the conjecture for . Conversely, assuming that the conjecture holds for , we get the following method to prove that the conjecture also holds for : we have to prove that monic binary forms can only satisfy

when . Suppose we have binary forms

satisfying the equation. If for each , then we get

and hence by the conjecture for . Otherwise we can assume, by permuting the and acting with , that . Now, we expand the sum

where the coefficients are polynomials in the and we compute the reduced Gröbner basis (with respect to some monomial ordering) of the ideal generated by the and the polynomial . If the conjecture holds for , then this Gröbner basis will be . And, if the Gröbner basis is , then cannot be , which means that and hence that the conjecture holds for .

We can now prove Theorem 1.6.

Proof of Theorem 1.6.

We consider the cases of the theorem separately.

  • The coefficients of the binary form

    at are, up to constant factors, power sums in the variables . These generate the invariant ring . This implies that

    and hence Conjecture 2 holds for . So (i) holds by Proposition 4.1.

  • Both Conjecture 2 and Shapiro’s Conjecture are trivial for . Assume that and let be a binary form whose coefficient at equals . Then for some (not necessarily monic) binary forms by [FOS12, Theorem 4]. Fix as the basis of and consider as a symmetric matrix. The linear map

    sends to . From this we see that the matrix has a as entry in its top-left corner. Now it follows from Proposition 5.3 that

    for some monic binary forms . So we have

    as desired.

  • The remaining cases are checked by computer, but we use one more observation: the system of homogeneous equations in the constructed in the inductive strategy given in Remark 4.2 has only integral coefficients and is homogeneous relative to the grading coming from the action of one-dimensional torus . Hence, we are checking whether a certain subvariety of a weighted projective space defined over  has no -points. To achieve this, it is enough to show that the subvariety has no -points for some prime . This allows us to work modulo some prime (e.g., the prime is enough), which makes the computation more efficient and lets it finish successfully. ∎

5. Minimal orbits

Let be a connected and reductive algebraic group over . Let be an irreducible rational representation of . Fix a Borel subgroup of and a maximal torus in . Let span the unique -stable one-dimensional subspace in , i.e., the highest weight space. Set . This is the affine cone over the homogeneous variety given by the orbit of . Let be the function that spans the unique -stable one-dimensional subspace of and is normalized so that . In this setting, we study Question 1.7, i.e., whether the maximal -rank of a vector in is also the maximal monic -rank of a vector in . As positive evidence, we treat the examples from Theorem 1.8.

5.1. Binary forms

Consider the case where

  • the vector space consists of binary forms of degree ;

  • the group acts on in the natural way;

  • the variety consists of powers of linear forms; and

  • the linear function sends a polynomial to its coefficient at .

Here the -rank is also called the Waring rank. Using the Apolarity Lemma (see, e.g., [IK99, Lemma 1.15]), one can show that has Waring rank and, in fact, the maximal Waring rank of a binary form of degree is exactly (see, e.g., [Rez13, Theorem 4.9]). By Theorem 1.6(i), the maximal monic rank with respect to equals as well. Hence, the answer to Question 1.7 is affirmative is this instance. Moreover, all open secant varieties of  are closed by Proposition 3.3—the coefficients of in the sum of -th powers of linear forms are the first power sums in and generate the invariant ring .

5.2. Rectangular matrices

Consider the case where

  • the vector space consists of matrices;

  • the group acts by left and right multiplication;

  • the variety consists of rank matrices; and

  • the linear function sends a matrix to its top-left entry.

Let be the affine space of matrices with and take .

Proposition 5.1.

We have

for all .

Proof.

The inclusions

are clear. Let be a matrix with . Our goal is to prove that . We prove this by induction on . Write . Then, by acting with the subgroup of -invariant elements of , we may assume that is the diagonal matrix with a  as its top-left entry followed by ones. If , then is the sum of copies of the matrix with just a as its top-left entry. Note that, in particular, this handles the case . Next, assume that . Then, from the fact that the equality

decomposes the matrix on the left as a sum of two matrices of rank with and as entries in their top-left corners, we see that there is a decomposition with and matrices such that . By induction, it follows that and hence . This concludes the proof. ∎

It follows from the proposition that the rank of a matrix in coincides with its monic rank. So in particular, the maximal monic rank is equal to the maximal rank.

5.3. Symmetric matrices

Consider the case where

  • the vector space

    consists of symmetric matrices;

  • the group acts by for and ;

  • the variety consists of rank matrices; and

  • the linear function sends a matrix to its top-left entry.

Let be the affine space of matrices with and take .

Remark 5.2.

The vector space can be viewed as the space of quadratic forms in the variables by associating the quadric

to a symmetric matrix . So, the variety corresponds to the set of squares of linear forms and affine space corresponds to the set of polynomials with coefficient at .

Proposition 5.3.

We have

for all .

Proof.

As in the proof of Proposition 5.1, it suffices to prove that every symmetric matrix with is an element of . We again first replace by a diagonal matrix: it is well-known that every symmetric matrix is congruent to a diagonal matrix. So we can write with and diagonal. By going though the proof of this fact, one can check that can be chosen so that its action on is -invariant and is the diagonal matrix with a  as its top-left entry followed by ones. This reduces the problem to the case where is this diagonal matrix. Now, from the fact that

we see that there is a decomposition with and such that . We again conclude that . ∎

Again, it follows from the proposition that the rank of a matrix in coincides with its monic rank. And in particular, the maximal monic rank is equal to the maximal rank.

5.4. tensors

Consider the case where

  • the vector space consists of tensors;

  • the group acts on in the natural way;

  • the variety