1. Introduction
Fix integers s≥2 and mi>0, 1≤i≤s, and an algebraically closed base field 𝕂. Let T∈⊗1≤i≤s𝕂mi+1 be a tensor of format (m1+1)×⋯×(ms+1) over 𝕂. The tensor rank r(T) of T is the minimal integer x≥0 such that T=∑i=1xv1,i⊗⋯⊗vs,i with vj,i∈𝕂mj+1 (see [1–6]). Classical papers (e.g., [7]) continue to suggest new results (see [8]). Let t(m1,…,ms) be the maximum of all integers r(T), T∈⊗1≤i≤s𝕂mi+1. In this paper we prove the following result.

Theorem 1.
For all integers s≥2 and m1≥⋯≥ms>0 one has t(m1,…,ms)≤∏i≠2(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 m1≫m2≫⋯≫ms>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 λ∈𝕂∖{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 ⊗1≤i≤s𝕂mi+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 Y⊂ℙn and any P∈〈Y〉 the Y-rank rY(P) of P is the minimal cardinality of a finite set A⊂Y such that P∈〈A〉. Now assume 〈Y〉=ℙn. The maximal Y-rank ρY is the maximum of all integers rY(P), P∈ℙn. Fix integers s>0, mi≥0, 1≤i≤s, and di>0, 1≤i≤s. Set T(m1,…,ms):=ℙm1×⋯×ℙms. Let νd1,…,ds:T(m1,…,ms)→ℙr, r:=-1+∏1≤i≤s(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 f∈𝕂[xi,j], 1≤i≤s, 0≤j≤mi, whose nonzero monomials have degree di with respect to the variables xi,j, 0≤j≤mi. 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 P∈ℙr, r:=-1+∏1≤i≤s(mi+1). Let {λT}λ∈𝕂∖{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, 1≤i≤s, and di, 1≤i≤s. A small box of T(m1,…,ms) is a closed set L1×⋯×Ls⊂T(m1,…,ms) with Li being a hyperplane of ℙmi for all i. A large box of T(m1,…,ms) is a product L1×⋯×Ls⊂T(m1,…,ms) such that there is j∈{1,…,s} with Lj⊂ℙmj being a hyperplane, while Li=ℙmi for all i≠j. 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) B⊂T(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, 1≤i≤s, and di, 1≤i≤s, and set r:=-1+∏1≤i≤s(mi+dimi). Fix P∈ℙr. The rank rm1,…,ms;d1,…,ds(P) of P is the minimal cardinality of a finite set A⊂T(m1,…,ms;d1,…,ds) such that P∈〈A〉. 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) B⊂T(m1,…,ms;d1,…,ds) there is a finite set A⊂T(m1,…,ms;d1,…,ds)∖B with P∈〈A〉 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)), 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 s≥2 and m1≥⋯≥ms>0 one has t′′(m1,…,ms)≤∏i≠2(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 [9] 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 s≥2 and mi>0, 1≤i≤s. Fix j∈{1,…,s} and let πj:T(m1,…,ms)→T(m1,…,mj-1,mj+1,…,ms) be the projection. For any small polybox B⊂T(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 m1≥m2>0. For each P∈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 E⊂ℙm1 of hyperplanes there is a set A⊂T(m1,m2)∖E×ℙm2 such that ♯(A)=t and P∈〈ν1,1(A)〉. Let t~(m1,m2) be the maximum of all integers t~m1,m2(P), 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 m1≥m2>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 P∈ℙr, r=m2+2m, and a union E⊂ℙm of finitely many hyperplanes. Fix v∈V⊗V inducing P and E′⊊V inducing E. Fix a basis e0,…,em of V such that ei∉E′ for all i. We may write v=∑i=0mei⊗wi for some wi∈V. 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 s≥3 and use induction on s. Fix P∈ℙr and a small polybox B⊂T(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 L⊂ℙms. Set T(m1,…,ms;s,L):=T(m1,…,ms-1)×L⊂T(m1,…,ms), E:=ν1,…,1(T(m1,…,ms;s,L)), F:=〈E〉, and R:=-1+∏1≤i≤s-1(mi+1). We have dim(F)=-1+ms∏1≤i≤s-1(mi+1). Let ℓ:ℙr∖F→ℙR denote the linear projection from F. Notice that F∩T(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 ms≥2, then we use induction on ms to apply Theorem 4 to E. Set ℓ':=ℓ∣T(m1,…,ms;1,…,1)∖E. Notice that ℓ' induces a surjection ℓ':T(m1,…,ms;1,…,1)∖E→T(m1,…,ms-1;1,…,1) (projection onto the first s-1 factors). Let B1 denote the closure of ℓ'(B∖B∩E) 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 B∩E is a small polybox of E. First assume P∈E. Since B∩E is a small polybox of E, the inductive assumption gives the existence of a set A⊂E∖A∩B such that P∈〈A〉 and ♯(A)≤ms×∏1≤i≤s-1,i≠2(mi+1). Hence rT(m1,…,ms;1,…,1)′′(P)<∏i≠2(mi+1). Now assume P∉F. Hence ℓ(P) is defined. Since B1 is a small polybox, there is B⊂T(m1,…,ms-1;1,…,1)∖B1 such that ℓ(P)∈〈B〉 and ♯(B)≤t′′(m1,…,ms-1)≤(m1+1)×∏3≤i≤s-1(mi+1). Since ℓ' is surjective, there is B2⊂E such that ℓ'(B2)=B. Since B2∩E=∅ and F∩T(m1,…,ms;1,…,1)=E, we have B2∩F=∅. Hence ℓ is defined at each point of B2. Since P∈〈B〉 and ℓ(B2)=B, there is O∈F such that P∈〈{O}∪B2〉. Since B∩E is a small polybox, there is B3⊂E∖B∩E such that O∈〈B3〉 and ♯(B3)≤t′′(m1,…,ms-1)≤ms×∏1≤i≤s,i≠2(mi+1). We have P∈〈B2∪B3〉 and ♯(B2∪B3)≤∏i≠2(mi+1).