Polynomials of minimal border rank

Technical Summary: Polynomials of Minimal Border Rank

Problem Statement
The paper addresses the classification of homogeneous polynomials of minimal border rank in $n$ variables and degree $d$. The Waring rank of a polynomial $f$ is the minimal number $r$ such that $f$ can be written as a sum of $r$ $d$-th powers of linear forms. The border rank, denoted $\text{brk}(f)$, is the minimal $r$ such that $f$ lies in the Zariski closure of the set of polynomials with Waring rank $r$. A polynomial is "concise" if it cannot be expressed in fewer than $n$ variables after a linear change of coordinates. Since a concise polynomial must have border rank at least $n$, those achieving this lower bound are said to have "minimal border rank."

While minimal border rank polynomials for $n \leq 5$ were previously classified by Landsberg and Teitler [LT10], the authors note that this classification contains inaccuracies. Furthermore, the behavior of these polynomials for $n \geq 6$ and specific low-degree cases ($d < n-1$) remained incomplete or unclear. The paper aims to provide a corrected and extended classification for $n \leq 7$ (and partially for $n=8,9$) under the condition $d \geq n-1$.

Methodology
The authors establish a correspondence between minimal border rank polynomials and specific algebraic structures, specifically smoothable Gorenstein algebras. The methodology relies on three key theoretical pillars:

1. Border Rank and Smoothable Rank: Building on [BB14], the authors utilize the fact that for $d \geq r-1$, a polynomial has border rank $r$ if and only if it lies in the linear span of a degree $r$ smoothable scheme embedded in the Veronese variety.
2. Cactus Apolarity and Gorenstein Algebras: Theorem 3.11 (a symmetric version of the cactus apolarity lemma from [JJ26]) states that concise tensors of minimal smoothable rank are precisely the evaluation tensors of Gorenstein algebras. Specifically, a concise tensor $f \in S^d V$ has symmetric border rank $n$ (where $\dim V = n$) if and only if $f$ is equivalent to the evaluation tensor $\mu^{(d-1)}_A$ of a smoothable $n$-dimensional Gorenstein algebra $A$.
3. Centroids and Algebra Recovery: To ensure a one-to-one correspondence between non-isomorphic algebras and non-equivalent polynomials, the authors employ the concept of the centroid of a tensor. Proposition 3.13 demonstrates that for $d \geq 3$, the centroid of the multiplication tensor of an algebra $A$ is isomorphic to $A$ itself. This allows the recovery of the algebra from the polynomial.

The authors leverage the classification of finite-dimensional algebras up to dimension 7 provided by Casnati [Cas10]. They develop an algorithm (Algorithm 4.3) to compute the iterated multiplication tensor (evaluation tensor) for a given local Gorenstein algebra. This algorithm is implemented in Macaulay2.

Key Contributions and Results

• Classification for $n \leq 7$: By combining the correspondence between minimal border rank polynomials and smoothable Gorenstein algebras with the classification in [Cas10], the authors classify all minimal border rank polynomials in $n \leq 7$ variables for $d \geq \max\{3, n-1\}$. The results are presented in Table 1, listing the ideals of the Gorenstein algebras and the corresponding polynomial forms.
• Correction of Previous Work: The authors apply their centroid-based verification algorithm to the classification in [LT10, Theorem 10.4] (for $n=4$). They demonstrate that two polynomials listed there do not satisfy the necessary conditions (specifically, they fail the centroid dimension test or are not concise), thereby correcting the existing literature.
• Infinite Families for $n \geq 8$: The paper establishes that for $n \geq 8$, there are infinitely many non-isomorphic smoothable Gorenstein algebras. Consequently, for any $n \geq 8$ and $d \geq 3$, there exist infinitely many nonequivalent polynomials of minimal border rank. The authors provide an explicit infinite family of such polynomials in 8 variables derived from the apolar algebras of plane cubics (Example 4.10).
• Algorithmic Verification: The paper provides a practical algorithm (Proposition 4.13) to verify whether a given concise polynomial has minimal border rank. The test checks if the polynomial is 1-generic (via a non-vanishing Hessian determinant) and if the dimension of its centroid is at least $n$.
• Handling Low Degrees: The authors note that for cases where $d < n-1$ (e.g., $S^3 \mathbb{C}^5$), the correspondence with smoothable algebras breaks down because border rank and smoothable rank may differ. They identify that additional minimal border rank polynomials exist in these regimes (e.g., the specific case in Remark 4.7) and leave the full classification of these remaining cases for future work.

Significance and Claims
The paper claims to resolve the classification of minimal border rank polynomials for $n \leq 7$ (with the exception of specific low-degree cases where smoothable and border ranks diverge) by reducing the geometric problem to the algebraic classification of Gorenstein algebras.

The authors emphasize that their approach corrects inaccuracies in the seminal work of Landsberg and Teitler [LT10] and extends the classification to $n=6$ and $n=7$. Furthermore, they highlight a counter-intuitive result regarding dimension counts: while one might expect a finite number of minimal border rank polynomials for fixed $n$ and $d$, the existence of infinite families of Gorenstein algebras for $n \geq 8$ implies an infinite number of nonequivalent minimal border rank polynomials.

The work is supported by supplementary material containing Macaulay2 code that implements the computation of evaluation tensors from Gorenstein algebras and the verification of minimal border rank for specific polynomials. The authors explicitly state that the classification for the remaining cases where $d < n-1$ (specifically $S^3 \mathbb{C}^5, S^3 \mathbb{C}^6, S^4 \mathbb{C}^6, S^3 \mathbb{C}^7, S^4 \mathbb{C}^7, S^5 \mathbb{C}^7$) is left for future research.