# On Comon’s and Strassen’s Conjectures

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## Abstract

**:**

## 1. Introduction

## 2. Notation

#### 2.1. Flattenings

**Remark**

**1.**

#### 2.2. Rank and Border Rank

## 3. Comon’s Conjecture

**Conjecture 1**(Comon’s).

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Proposition**

**2.**

**Proof.**

**Remark**

**2.**

## 4. Strassen’s Conjecture

**Conjecture 2**(Strassen’s).

**Proposition**

**3.**

**Proof.**

**Remark**

**3.**

**Proposition**

**4.**

**Proof.**

**Proposition**

**5.**

## 5. On the Rank of ${\mathit{x}}_{\mathbf{0}}\mathit{F}$

**Remark**

**4.**

**Proposition**

**6.**

**Proof.**

**Definition**

**1.**

**Corollary**

**1.**

**Proof.**

- i
- $(n,d)=(2,2)$. The variety $\mathbb{S}e{c}_{3}({\mathcal{V}}_{3}^{2})$ is the hypersurface in ${\mathbb{P}}^{9}$ cut out by the Aronhold invariant, see for instance (Section 1.1 in [4]). With a Macaulay2 computation we prove that if $F\in \mathbb{S}e{c}_{2}({\mathcal{V}}_{2}^{2})$ is general then the Aronhold invariant does not vanish at ${x}_{0}F$, hence $rank{x}_{0}F=2rankF$.
- ii
- $(n,d)=(2,3)$. The varieties $\mathbb{S}e{c}_{5}({\mathcal{V}}_{4}^{2})$ and $\mathbb{S}e{c}_{3}({\mathcal{V}}_{3}^{2})$ are both hypersurfaces, given respectively by the determinant of the catalecticant matrix of second partial derivatives and the Aronhold invariant (Section 1.1 in [4]). With Macaulay2 we prove that the determinant of the second catalecticant matrix does not vanish at ${x}_{0}F$ for $F\in \mathbb{S}e{c}_{3}({\mathcal{V}}_{3}^{2})$ general, hence $rank{x}_{0}F=2rankF$.
- iii
- $(n,d)=(3,3)$. The secant variety $\mathbb{S}e{c}_{9}({\mathcal{V}}_{4}^{3})$ is the hypersurface cut out by the second catalecticant matrix (Section 1.1 in [4]) while $\mathbb{S}e{c}_{5}({\mathcal{V}}_{3}^{3})$ is the entire osculating space. A Macaulay2 computation shows that ${\mathbb{T}}_{[{x}_{0}^{4}]}^{3}{\mathcal{V}}_{4}^{3}\subseteq \mathbb{S}e{c}_{9}({\mathcal{V}}_{4}^{3})$. This proves that $rank{x}_{0}F<2rankF$, for F general.
- iv
- $(n,d)=(4,3)$. In this case $\mathbb{S}e{c}_{8}({\mathcal{V}}_{3}^{4})={\mathbb{T}}_{[{x}_{0}^{4}]}^{3}{\mathcal{V}}_{4}^{4}$ and $\mathbb{S}e{c}_{14}({\mathcal{V}}_{4}^{4})$ is given by the determinant of the second catalecticant matrix (Section 1.1 in [4]). Again using Macaulay2 we show that ${\mathbb{T}}_{[{x}_{0}^{4}]}^{3}{\mathcal{V}}_{4}^{4}\subseteq \mathbb{S}e{c}_{14}({\mathcal{V}}_{4}^{4})$. This proves that $rank{x}_{0}F<2rankF$, for F general.

**Corollary**

**2.**

**Proof.**

#### Macaulay2 Implementation

`Comon-1.0.m2`we provide a function called

`Comon`which operates as follows:

- -
`Comon`takes in input three natural numbers $n,d,h$;- -
- if $h<\left(\genfrac{}{}{0pt}{}{n+\lfloor \frac{d}{2}\rfloor}{n}\right)$ then the function returns that Comon’s conjecture holds for the general degree d polynomial in $n+1$ variables of rank h by the usual flattenings method in Proposition 1. If not, and d is even then it returns that the method does not apply;
- -
- if d is odd and $\left(\genfrac{}{}{0pt}{}{n+k}{n}\right)<2\left(\genfrac{}{}{0pt}{}{n+k-1}{n}\right)$, where $k=\lfloor \frac{d+1}{2}\rfloor $, then again it returns that the method does not apply;
- -
- if d is odd, $\left(\genfrac{}{}{0pt}{}{n+k}{n}\right)\ge 2\left(\genfrac{}{}{0pt}{}{n+k-1}{n}\right)$ and $2h-1>\left(\genfrac{}{}{0pt}{}{n+k}{n}\right)$ then it returns that the method does not apply since $2h-2$ must be smaller than the number of order k partial derivatives;
- -
- if d is odd, $\left(\genfrac{}{}{0pt}{}{n+k}{n}\right)\ge 2\left(\genfrac{}{}{0pt}{}{n+k-1}{n}\right)$ and $2h-1\le \left(\genfrac{}{}{0pt}{}{n+k}{n}\right)$ then
`Comon`, in the spirit of Remark 4, produces a polynomial of the form$$F=\sum _{i=1}^{h}{({a}_{i,0}{x}_{0}+\cdots +{a}_{i,n}{x}_{n})}^{d}$$ - -
- if $det(P)=0$ then
`Comon`returns that the method does not apply, otherwise it returns that Comon’s conjecture holds for the general degree d polynomial in $n+1$ variables of rank h.

`random`is involved

`Comon`may return that the method does not apply even though it does. Clearly, this event is extremely unlikely. Thanks to this function we are able to prove that Comon’s conjecture holds in some new cases that are not covered by Proposition 1. Since the case $n=1$ is covered by Proposition 6 in the following we assume that $n\ge 2$.

**Theorem**

**1.**

- -
- $d=3$ and $2\le n\le 30$;
- -
- $d=5$ and $3\le n\le 8$;
- -
- $d=7$ and $n=4$.

**Proof.**

`Comon`exactly as shown in Example 1 below. □

**Example**

**1.**

`Comon`in a few interesting cases:

`Macaulay2, version 1.12`

`with packages: ConwayPolynomials, Elimination, IntegralClosure, InverseSystems,`

`LLLBases, PrimaryDecomposition, ReesAlgebra, TangentCone`

`i1 : loadPackage "Comon-1.0.m2";`

`i2 : Comon(5,3,4)`

`Lowest rank for which the usual flattenings method does not work = 6`

`o2 = Comon’s conjecture holds for the general degree 3 homogeneous polynomial`

`in 6 variables of rank 4 by the usual flattenings method`

`i3 : Comon(5,3,6)`

`Lowest rank for which the usual flattenings method does not work = 6`

`o3 = Comon’s conjecture holds for the general degree 3 homogeneous polynomial`

`in 6 variables of rank 6`

`i4 : Comon(5,3,7)`

`Lowest rank for which the usual flattenings method does not work = 6`

`o4 = The method does not apply --- The determinant vanishes`

`i5 : Comon(5,5,21)`

`Lowest rank for which the usual flattenings method does not work = 21`

`o5 = Comon’s conjecture holds for the general degree 5 homogeneous polynomial`

`in 6 variables of rank 21`

`i6 : Comon(4,7,35)`

`Lowest rank for which the usual flattenings method does not work = 35`

`o6 = Comon’s conjecture holds for the general degree 7 homogeneous polynomial`

`in 5 variables of rank 35`

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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Casarotti, A.; Massarenti, A.; Mella, M. On Comon’s and Strassen’s Conjectures. *Mathematics* **2018**, *6*, 217.
https://doi.org/10.3390/math6110217

**AMA Style**

Casarotti A, Massarenti A, Mella M. On Comon’s and Strassen’s Conjectures. *Mathematics*. 2018; 6(11):217.
https://doi.org/10.3390/math6110217

**Chicago/Turabian Style**

Casarotti, Alex, Alex Massarenti, and Massimiliano Mella. 2018. "On Comon’s and Strassen’s Conjectures" *Mathematics* 6, no. 11: 217.
https://doi.org/10.3390/math6110217