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Waring rank of determinant

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Statement

Definitions and mathematical statement

Fix an integer $d\ge1$ and work in the polynomial ring $R=\mathbb C[x_{ij}:1\le i,j\le d]$, whose $d^2$ variables are independent. The generic determinant is the homogeneous degree-$d$ polynomial \[ D_d=\det(x_{ij})= \sum_{\sigma\in S_d}\operatorname{sgn}(\sigma) \prod_{i=1}^d x_{i,\sigma(i)}, \] where $S_d$ is the group of permutations of $\{1,\ldots,d\}$. A homogeneous linear form is an expression $\ell=\sum_{i,j}a_{ij}x_{ij}$ with coefficients $a_{ij}\in\mathbb C$ and no constant term.

The complex Waring rank of a nonzero homogeneous degree-$d$ form $F$ is \[ R_{\mathrm W}(F)=\min\left\{r\ge1: F=\sum_{\nu=1}^r\ell_\nu^d\text{ identically in }R\right\}. \] Allowing scalar coefficients in front of the powers gives the same number, because each complex scalar has a $d$th root and can be absorbed into its linear form. The forms may involve all entries of the matrix and need not satisfy any symmetry or use disjoint variables.

Determine $R_{\mathrm W}(D_d)$ as a function of $d$. An exact answer consists of a power-sum expression achieving the proposed value and a lower bound excluding every expression with fewer summands. The equality is polynomial identity on all matrices, including singular matrices. It is not enough to express the determinant as products of unrelated linear forms, or as a limit of shorter power sums: these concern other rank notions. In particular, border rank and unsymmetrized tensor rank are not the invariant asked for.

Short English statement

What is the smallest number of complex $d$th powers of linear forms whose sum is the generic $d$ by $d$ determinant?

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Math-1009-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

gpt 6 astra ultra attempt on Math-1009

Waring rank of determinant

Definitions and mathematical statement

Fix an integer $d\ge1$ and work in the polynomial ring $R=\mathbb C[x_{ij}:1\le i,j\le d]$, whose $d^2$ variables are independent. The generic determinant is the homogeneous degree-$d$ polynomial \[ D_d=\det(x_{ij})= \sum_{\sigma\in S_d}\operatorname{sgn}(\sigma) \prod_{i=1}^d x_{i,\sigma(i)}, \] where $S_d$ is the group of permutations of $\{1,\ldots,d\}$. A homogeneous linear form is an expression $\ell=\sum_{i,j}a_{ij}x_{ij}$ with coefficients $a_{ij}\in\mathbb C$ and no constant term.

The complex Waring rank of a nonzero homogeneous degree-$d$ form $F$ is \[ R_{\mathrm W}(F)=\min\left\{r\ge1: F=\sum_{\nu=1}^r\ell_\nu^d\text{ identically in }R\right\}. \] Allowing scalar coefficients in front of the powers gives the same number, because each complex scalar has a $d$th root and can be absorbed into its linear form. The forms may involve all entries of the matrix and need not satisfy any symmetry or use disjoint variables.

Determine $R_{\mathrm W}(D_d)$ as a function of $d$. An exact answer consists of a power-sum expression achieving the proposed value and a lower bound excluding every expression with fewer summands. The equality is polynomial identity on all matrices, including singular matrices. It is not enough to express the determinant as products of unrelated linear forms, or as a limit of shorter power sums: these concern other rank notions. In particular, border rank and unsymmetrized tensor rank are not the invariant asked for.

Short English statement

What is the smallest number of complex $d$th powers of linear forms whose sum is the generic $d$ by $d$ determinant?

Research attempt

Scope and process.
This is a GPT-6 Astra Ultra research attempt dated 1 October 2026, using primary-source browsing and elementary mathematical checks. The canonical definition above is retained. Results below are restricted results or reductions, not a claim of a new solution to the entire catalogue question. No formal proof assistant or external mathematical referee checked this text.
Literature and conventions.
The rank is complex Waring rank, with literal sums of powers as in the definition. The primary paper [R1] develops stronger lower bounds for determinants using invariant forms. We use only derivative spaces and polarization, proved below; the resulting estimates are benchmarks and are not advertised as the best known bounds.
Derivative-space lower bound.
For a degree-$d$ polynomial $F$, let $V_k(F)$ be the span of all its partial derivatives of total order $k$, where $0\le k\le d$. If $F=\sum_{\nu=1}^r\ell_\nu^d$, every such derivative is a scalar multiple of some $\ell_\nu^{d-k}$, summed over $\nu$. Hence $\dim V_k(F)\le r$. This uses the actual decomposition and gives a lower bound for the rank requested in the problem.

For $D_d$, a nonzero order-$k$ derivative differentiates distinct rows and distinct columns, and yields, up to sign, a complementary $(d-k)$-minor. Every such minor occurs: pair the omitted rows and columns in any order and differentiate in those $k$ entries. Derivatives repeating a variable, row or column vanish by the multilinearity of each determinant monomial. Distinct minors of the same size are linearly independent, because each monomial determines its row set and column set and hence occurs in just one of these minors. This remains valid for the unique empty minor when $k=d$. Consequently \[ \dim V_k(D_d)=\binom dk^2,\qquad R_{\mathrm W}(D_d)\ge\binom d{\lfloor d/2\rfloor}^{\!2}. \]

An explicit general upper bound.
For independent variables $y_1,\ldots,y_d$, character cancellation on the sign cube gives \[ y_1\cdots y_d= \frac{1}{2^d d!}\sum_{\epsilon\in\{-1,1\}^d} \left(\prod_i\epsilon_i\right) \left(\sum_i\epsilon_i y_i\right)^d. \] To check this, expand the powers. The sign sum of a monomial with exponents $a_i$ is zero unless every $a_i$ is odd. Since their sum is $d$, this forces every $a_i=1$. Its multinomial coefficient is $d!$, so the surviving coefficient is one. The summands indexed by $\epsilon$ and $-\epsilon$ are identical, including the prefactor, so only $2^{d-1}$ powers are needed. Apply this separately to the $d!$ signed monomials in the determinant and absorb each nonzero complex coefficient into a $d$th root. This proves \[ R_{\mathrm W}(D_d)\le d!\,2^{d-1}. \] Possible cancellations between these powers can only improve this bound.
Exact low-dimensional checks.
For $d=1$ the rank is one. For $d=2$, write the entries as $a,b,c,e$; then \[ ae-bc=\frac14\left((a+e)^2-(a-e)^2 -(b+c)^2+(b-c)^2\right). \] This is a four-square complex decomposition after absorbing signs and the factor $1/4$. The first derivatives are $e,-c,-b,a$, which are linearly independent, so fewer than four powers are impossible. Thus $R_{\mathrm W}(D_2)=4$. Over $\mathbb R$, requiring sums of squares with no signed coefficients would be a different problem, so the complex-field hypothesis is essential to this verification.
Where the attack stops.
Already for $d=3$ these elementary arguments give only $9\le R\le24$. The derivative dimension records the span generated by potential powers, but does not show that a spanning collection can actually realize the determinant with that many powers. Conversely, treating permutation monomials independently wastes cancellations. Neither equality in the lower bound nor optimality of the upper construction has been proved. Higher-dimensional apolar or representation-theoretic restrictions, such as those studied in [R1], are additional information, not a consequence of the displayed dimensions.
Final status.
UNRESOLVED as a function of arbitrary $d$. The exact values for $d=1,2$ and general elementary two-sided bounds are established; no new exact higher-dimensional rank is claimed.

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