GEOMETRY ISRN Geometry 2090-6315 Hindawi Publishing Corporation 241835 10.1155/2013/241835 241835 Research Article An Upper Bound for the Tensor Rank Ballico E. Cieśliński J. L. Montaldi J. Porti J. Department of Mathematics University of Trento 38123 Povo Italy unitn.it 2013 23 5 2013 2013 18 04 2013 12 05 2013 2013 Copyright © 2013 E. Ballico. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Let T be a tensor of format m1+1××ms+1, m1ms>0, over . We prove that T has tensor rank at most i2mi+1.

1. Introduction

Fix integers s2 and mi>0, 1is, and an algebraically closed base field 𝕂. Let T1is𝕂mi+1 be a tensor of format (m1+1)××(ms+1) over 𝕂. The tensor rank r(T) of T is the minimal integer x0 such that T=i=1xv1,ivs,i with vj,i𝕂mj+1 (see ). Classical papers (e.g., ) continue to suggest new results (see ). Let t(m1,,ms) be the maximum of all integers r(T), T1is𝕂mi+1. In this paper we prove the following result.

Theorem 1.

For all integers s2 and m1ms>0 one has t(m1,,ms)i2(mi+1).

This result is not optimal. It is not sharp when s=2, since t(m1,m2)=m2+1 by elementary linear algebra. For large s the bound should be even worse. In our opinion to get stronger results one should split the set of all (s;m1,,ms) into subregions. For instance, we think that for large s the cases with m1m2ms>0 and the cases with m1==ms are quite different.

We make the definitions in the general setting of the Segre-Veronese embeddings of projective spaces (i.e., of partially symmetric tensors), but we only use the case of the usual Segre embedding, that is, the usual tensor rank. The tensor T=0 has zero as its tensor rank. If λ𝕂{0}, then the tensors T and λT have the same rank. Hence it is sufficient to study the function “tensor rank” on the projectivisation of the vector space 1is𝕂mi+1. We may translate the tensor rank and the integer t(m1,,ms) in the following language. For each subset A of a projective space, let A denote the linear span of A. For each integral variety Yn and any PY the Y-rank rY(P) of P is the minimal cardinality of a finite set AY such that PA. Now assume Y=n. The maximal Y-rank ρY is the maximum of all integers rY(P), Pn. Fix integers s>0, mi0, 1is, and di>0, 1is. Set T(m1,,ms):=m1××ms. Let νd1,,ds:T(m1,,ms)r, r:=-1+1is(mi+dimi) be the Segre-Veronese embedding of multidegree (d1,,ds), that is, the embedding of T(m1,,ms) induced by the 𝕂-vector space of all polynomials f𝕂[xi,j], 1is, 0jmi, whose nonzero monomials have degree di with respect to the variables xi,j, 0jmi. Set T(m1,,ms;d1,,ds):=νd1,,ds(T(m1,,ms)). The variety T(m1,,ms;1,,1) is the Segre embedding of T(m1,,ms). Fix Pr, r:=-1+1is(mi+1). Let {λT}λ𝕂{0} be the set of all nonzero tensors of format (m1+1)××(ms+1) associated with P. We have r(T)=rT(m1,,ms;1,,1)(P). Hence t(m1,,ms)=ρT(m1,,ms;1,,1). To prove Theorem 1 we refine the notion of Y-rank in the following way.

Definition 2.

Fix positive integers s, mi, 1is, and di, 1is. A small box of T(m1,,ms) is a closed set L1××LsT(m1,,ms) with Li being a hyperplane of mi for all i. A large box of T(m1,,ms) is a product L1××LsT(m1,,ms) such that there is j{1,,s} with Ljmj being a hyperplane, while Li=mi for all ij. A small polybox (resp., large polybox) of T(m1,,ms) is a finite union of small (resp., large) boxes of T(m1,,ms). A small box (resp., small polybox, resp., large box, resp., large polybox) BT(m1,,ms;d1,,ds) is the image by νd1,,ds of a small box (resp., small polybox, resp., large box, resp., large polybox) of T(m1,,ms).

Definition 3.

Fix positive integers s, mi, 1is, and di, 1is, and set r:=-1+1is(mi+dimi). Fix Pr. The rank rm1,,ms;d1,,ds(P) of P is the minimal cardinality of a finite set AT(m1,,ms;d1,,ds) such that PA. The unboxed rank (resp., small unboxed rank) rm1,,ms;d1,,ds(P) (resp., rm1,,ms;d1,,ds(P)) of P is the minimal integer t>0 such that for each large polybox (resp., small polybox) BT(m1,,ms;d1,,ds) there is a finite set AT(m1,,ms;d1,,ds)B with PA and (A)=t. Let t(m1,,ms;d1,,ds) (resp., t(m1,,ms;d1,,ds), resp., t(m1,,ms;d1,,ds)) be the maximum of all integers rm1,,ms;d1,,ds(P) (resp., rm1,,ms;d1,,ds(P), resp., rm1,,ms;d1,,ds(P)), Pr.

Notice that t(m1,,ms)=t(m1,,ms;1,,1).

Since rm1,,ms;d1,,ds(P)rm1,,ms;d1,,ds′′(P)rm1,,ms;d1,,ds(P) for all P, we have t(m1,,ms;d1,,ds)t′′(m1,,ms;d1,,ds). Hence Theorem 1 is an immediate corollary of the following result.

Theorem 4.

For all integers s2 and m1ms>0 one has t′′(m1,,ms)i2(mi+1).

We hope that the definitions of unboxed rank and small unboxed rank are interesting in themselves, not just as a tool. As far as we know the best upper bound for the symmetric tensor rank is due to Białynicki-Birula and Schinzel ([9, 10]). In  Białynicki-Birula and Schinzel used the corresponding notion in the case s=1.

2. Proof of Theorem <xref ref-type="statement" rid="thm2">4</xref> Remark 5.

Fix integers s2 and mi>0, 1is. Fix j{1,,s} and let πj:T(m1,,ms)T(m1,,mj-1,mj+1,,ms) be the projection. For any small polybox BT(m1,,ms) the set πj(B) is a small polybox of the Segre variety T(m1,mj-1,mj+1,,ms).

In the case s=2 we also need the following notation. Fix integers m1m2>0. For each Pr, r=(m1+1)(m2+1)-1, let t~m1,m2(P) be the minimal integer t>0 with the following property: for each finite union Em1 of hyperplanes there is a set AT(m1,m2)E×m2 such that (A)=t and Pν1,1(A). Let t~(m1,m2) be the maximum of all integers t~m1,m2(P), Pr. Obviously t(m1,m2)t′′(m1,m2)t~(m1,m2)t(m1,m2). Linear algebra gives t(m1,m2)=1+min{m1,m2}=m2+1.

Lemma 6.

For all integers m1m2>0 one has t′′(m1,m2)t~(m1,m2)m1+1.

Proof.

It is sufficient to prove the inequality t~(m1,m2)m1+1. Without losing generality we may assume m1=m2. Set m:=m1 and V:=𝕂m+1. Fix Pr, r=m2+2m, and a union Em of finitely many hyperplanes. Fix vVV inducing P and EV inducing E. Fix a basis e0,,em of V such that eiE for all i. We may write v=i=0meiwi for some wiV. Hence t~m,m(P)m+1.

Proof of Theorem <xref ref-type="statement" rid="thm2">4</xref>.

Lemma 6 gives the case s=2. Hence we may assume s3 and use induction on s. Fix Pr and a small polybox BT(m1,,ms;1,,1). For a fixed integer s we also use induction on ms, starting from the case ms=0 (in which we use s-1 instead of s).

Take a general hyperplane Lms. Set T(m1,,ms;s,L):=T(m1,,ms-1)×LT(m1,,ms), E:=ν1,,1(T(m1,,ms;s,L)),  F:=E, and R:=-1+1is-1(mi+1). We have dim(F)=-1+ms1is-1(mi+1). Let :rFR denote the linear projection from F. Notice that FT(m1,,ms;1,,1)=E. If ms=1, then we have E=T(m1,,ms-1;1,,1) and hence we use induction on s to apply Theorem 4 to E. If ms2, then we use induction on ms to apply Theorem 4 to E. Set ':=T(m1,,ms;1,,1)E. Notice that ' induces a surjection ':T(m1,,ms;1,,1)ET(m1,,ms-1;1,,1) (projection onto the first s-1 factors). Let B1 denote the closure of '(BBE) in T(m1,,ms-1;1,,1). Since B is a small polybox, B1 is a small polybox (Remark 5). For general L we may also assume that BE is a small polybox of E. First assume PE. Since BE is a small polybox of E, the inductive assumption gives the existence of a set AEAB such that PA and (A)ms×1is-1,i2(mi+1). Hence rT(m1,,ms;1,,1)′′(P)<i2(mi+1). Now assume PF. Hence (P) is defined. Since B1 is a small polybox, there is BT(m1,,ms-1;1,,1)B1 such that (P)B and (B)t′′(m1,,ms-1)(m1+1)×3is-1(mi+1). Since ' is surjective, there is B2E such that '(B2)=B. Since B2E= and FT(m1,,ms;1,,1)=E, we have B2F=. Hence is defined at each point of B2. Since PB and (B2)=B, there is OF such that P{O}B2. Since BE is a small polybox, there is B3EBE such that OB3 and (B3)t′′(m1,,ms-1)ms×1is,i2(mi+1). We have PB2B3 and (B2B3)i2(mi+1).

Acknowledgments

The author was partially supported by MIUR and GNSAGA of INdAM (Italy).

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