Border Affine Chow Rank in Homogeneous Forms

• Border affine Chow rank is defined as the minimal number of summands required for a homogeneous form to lie in the affine cone closure of decomposable forms.
• It leverages geometric tools such as secant varieties and Terracini’s lemma to assess decomposition thresholds and verify nondefectivity in generic cases.
• Under nondefective conditions, border affine Chow rank aligns with both Chow rank and affine Chow rank, with exceptions arising in specific quadratic defective cases.

Searching arXiv for papers on Chow rank, border Chow rank, and affine Chow rank. arXiv search query: "Chow rank secant variety affine Chow rank border Chow rank" I’m going to query arXiv for relevant papers to ground the article. Border affine Chow rank is the affine-closure analogue of Chow rank for homogeneous forms decomposed as sums of products of linear forms. In the projective setting, the Chow rank of a degree-$d$ form $f$ in $n+1$ variables is the minimal $s$ such that

$f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$

with each $\ell_{i,j}$ linear (Torrance, 2015). The corresponding border notion replaces exact decomposition by membership in the Zariski closure of the locus of such sums, while the affine variant transfers the same question from projective classes to affine cones. The cited literature treats Chow rank and border Chow rank explicitly and discusses affine Chow rank and border affine Chow rank only indirectly; accordingly, the affine-border notion is best understood as an extension suggested by the secant-variety formalism rather than as an independently developed theory in these papers (Torrance, 2015, Torrance et al., 2020).

1. Definition through decompositions and closure

For a homogeneous polynomial of degree $d$ in $n+1$ variables, the Chow rank is the smallest integer $s$ for which the polynomial can be written as a sum of $s$ products of $f$0 linear forms (Torrance, 2015). This definition places Chow rank among additive decomposition invariants, but with decomposable summands of the form $f$1 rather than pure powers.

The border Chow rank is the least $f$2 such that $f$3 lies in the Zariski closure of sums of $f$4 decomposable forms. In geometric language, this is the least $f$5 for which the projective class of $f$6 belongs to the $f$7th secant variety of the Chow variety (Torrance et al., 2020). The papers further state that affine Chow rank and border affine Chow rank refer to the minimal number of summands when decomposition is allowed in the affine, rather than projective, setting and potentially in the closure (Torrance et al., 2020).

Because the affine version is not developed in depth, several statements about border affine Chow rank are necessarily interpretive. The literature explicitly says that the affine variants are “not deeply discussed” and that “the identical argument applies so long as one works in affine cones, since the underlying linear geometry and counts remain valid” (Torrance et al., 2020). This suggests that border affine Chow rank is intended to be the affine-cone counterpart of border Chow rank, with no change in generic counting behavior when the passage between projective and affine settings is benign.

2. Geometric formulation via the Chow variety

The relevant projective variety is the Chow variety, denoted in one source by $f$8 and in another by $f$9, consisting of classes of completely reducible degree-$n+1$0 forms (Torrance, 2015, Torrance et al., 2020). Its secant varieties organize sums of decomposable forms: the $n+1$1th secant variety is the closure of the union of all $n+1$2-planes spanned by $n+1$3 points of the Chow variety (Torrance et al., 2020).

For generic forms, Chow rank is governed by the smallest secant variety that fills the ambient space. One paper states this as

$n+1$4

for a generic form (Torrance, 2015). The expected dimension of the $n+1$5th secant variety is

$n+1$6

(Torrance, 2015).

This secant-variety description is the basis for the border notions. Since secant varieties are Zariski-closed, border Chow rank is measured by secant membership rather than exact representability. A plausible implication for the affine-border setting is that one passes from projective secant varieties to their affine cones, preserving the same threshold $n+1$7 for generic forms whenever projectivization does not introduce exceptional behavior.

3. Expected dimension, defectivity, and generic rank

A secant variety is nondefective when its actual dimension equals the expected dimension, and defective when the dimension is smaller (Torrance, 2015). This distinction is central because generic Chow rank and generic border Chow rank coincide whenever the relevant secant variety fills the ambient space as expected.

The principal conjectural picture recorded in the literature is that $n+1$8 is nondefective unless $n+1$9 and $s$0 (Torrance, 2015). The quadratic cases are completely catalogued: $s$1 is defective if and only if $s$2 (Torrance, 2015). Outside these exceptions, the paper establishes nondefectivity for low secant order, specifically: if $s$3, then $s$4 is nondefective for all $s$5, unless $s$6 and $s$7 (Torrance, 2015).

Under nondefectivity, the generic Chow rank is given by the parameter count

$s$8

for the cases covered in the paper (Torrance, 2015). The same source states that, since the secant variety is Zariski-closed, the border Chow rank of a generic form coincides with its Chow rank, except possibly for defective cases (Torrance, 2015). It further notes that the affine version “would not alter the count for generic forms” (Torrance, 2015). Thus, for generic forms in the nondefective range, Chow rank, border Chow rank, and affine Chow rank are reported to coincide, and the same conclusion is presented in the paper’s summary as extending to affine Chow rank for generic forms (Torrance, 2015).

4. Established nondefectivity regimes and their implications

The strongest concrete nondefectivity statements in the supplied literature concern two families: low secant order in arbitrary degree and dimension, and all secant varieties for cubics and quaternary forms.

The following table compiles the explicit regimes stated in the papers.

Setting	Statement	Consequence
$s$9, $f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$0	defective	generic rank exceeds naive parameter count
$f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$1, $f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$2	nondefective	generic Chow and border Chow ranks follow expected count
cubics ($f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$3)	all secant varieties nondefective	generic Chow and border Chow ranks match parameter count
quaternary forms ($f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$4)	all secant varieties nondefective	generic Chow and border Chow ranks match parameter count

For cubics and quaternary forms, all secant varieties of the Chow variety are proved nondefective (Torrance et al., 2020). The paper states that this determines the Chow rank of generic cubics and quaternary forms by proving nondefectivity of all involved secant varieties (Torrance et al., 2020). It also states that, since nondefectivity holds generically, the border Chow rank and generic Chow rank coincide, and adds that the affine variants are expected to follow the same argument on affine cones (Torrance et al., 2020).

The same source gives generic-rank formulas for these two families: $f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$5 and

$f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$6

These formulas are presented as consequences of complete nondefectivity in those settings (Torrance et al., 2020). In the context of border affine Chow rank, the natural reading is that the same generic thresholds should govern the affine-border invariant whenever the affine-cone formulation is used without additional restrictions.

5. Methods: Terracini’s lemma, induction, and computation

The technical engine throughout this literature is Terracini’s lemma. For Chow varieties, one paper records the tangent-space computation in the form

$f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$7

for generic linear forms $f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$8 (Torrance, 2015). Another states the general version

$f=\sum_{i=1}^s \ell_{i,1}\cdots \ell_{i,d},$9

for generic $\ell_{i,j}$0 and generic $\ell_{i,j}$1 (Torrance et al., 2020). In both formulations, secant-dimension problems reduce to linear algebra on sums of tangent spaces.

The 2015 paper uses an induction method in $\ell_{i,j}$2: if nondefectivity is known at some base $\ell_{i,j}$3, then it is propagated to all larger $\ell_{i,j}$4 (Torrance, 2015). It also reports exhaustive computational checks with Macaulay2 for small $\ell_{i,j}$5, $\ell_{i,j}$6, and $\ell_{i,j}$7, confirming nondefectivity up to $\ell_{i,j}$8, except for the known defective quadratic cases (Torrance, 2015).

The 2020 paper generalizes a Brambilla–Ottaviani technique by employing Terracini’s lemma and Newton’s backward difference formula to compute dimensions of secant varieties of arbitrary projective varieties (Torrance et al., 2020). Its proof reduces nondefectivity to a finite set of base cases settled by a computer-assisted proof. The largest base case required “consisted of computing the dimension of a vector space constructed from the $\ell_{i,j}$9th secant variety of a degree-$d$0 Chow variety embedded in $d$1” (Torrance et al., 2020). The paper further describes explicit matrix constructions, computation of ranks over a large finite field, and the use of Eigen and FFLAS-FFPACK, together with publicly available certificates (Torrance et al., 2020).

For border affine Chow rank, these methods matter because they do not depend on an explicit exact decomposition algorithm; instead, they characterize closure membership geometrically. This suggests that affine-border questions for generic forms are governed primarily by secant-dimension and tangent-space analysis.

6. Relation to other rank notions and the role of “border”

Chow rank belongs to the broader landscape of tensor and polynomial ranks, especially Waring rank and border rank. The papers explicitly frame Chow rank as analogous to the theory for Waring ranks (Torrance, 2015). The distinction is that Waring decompositions use powers of linear forms, whereas Chow decompositions use products of possibly distinct linear forms.

The supplied literature on real symmetric tensors illustrates why border notions can diverge sharply from exact rank notions over non-algebraically closed fields. For fixed complex border rank in the Veronese setting, the real locus can decompose into manifolds on which the real rank is constant, and multiple typical real ranks can occur (Ballico, 2013). For example, for $d$2 and suitable $d$3, the typical real ranks in $d$4 are classified explicitly; more generally, for $d$5, $d$6, $d$7, the first two typical real ranks are $d$8 and $d$9 (Ballico, 2013).

That paper does not address Chow rank or affine Chow rank directly (Ballico, 2013). However, it provides a useful caution against conflating complex generic border behavior with real exact rank behavior. This suggests that, although generic border Chow rank and generic Chow rank coincide in the nondefective complex-projective regimes described above, analogous statements over $n+1$0 or for nongeneric forms may require additional stratified analysis rather than a single generic count.

7. Scope, limitations, and current interpretation of border affine Chow rank

The principal limitation is terminological and expository rather than geometric. The two Chow-variety papers explicitly develop Chow rank, secant varieties of the Chow variety, defectivity, and border Chow rank, but they do not provide an extensive standalone theory of affine Chow rank or border affine Chow rank (Torrance, 2015, Torrance et al., 2020). Instead, they state that the affine variant is not discussed in depth and that the affine setting should not alter the count for generic forms, because the same linear-geometric arguments apply on affine cones (Torrance, 2015, Torrance et al., 2020).

Accordingly, the most defensible encyclopedic formulation is the following. Border affine Chow rank is the least $n+1$1 such that a homogeneous form lies in the affine-cone closure of sums of $n+1$2 completely decomposable degree-$n+1$3 forms. In the regimes where secant varieties of the Chow variety are proved nondefective, the literature indicates that, for generic forms, Chow rank, border Chow rank, and affine Chow rank coincide, and the same reasoning plausibly extends to border affine Chow rank (Torrance, 2015, Torrance et al., 2020). The principal confirmed exceptions arise in the quadratic defective cases $n+1$4, $n+1$5 (Torrance, 2015).

A common misconception is that “border” always changes the generic answer. In the Chow-variety setting studied here, the cited papers indicate the opposite for generic forms in nondefective cases: the closure does not reduce the generic threshold $n+1$6 (Torrance, 2015, Torrance et al., 2020). Another misconception is that the affine version should produce substantially different generic counts from the projective one. The papers explicitly suggest that it does not, at least for generic forms and within the secant-variety framework (Torrance, 2015, Torrance et al., 2020).

Within current arXiv-documented treatment, border affine Chow rank is therefore best viewed as a derived notion anchored in secant varieties of the Chow variety, rigorously controlled for generic forms by nondefectivity results, but not yet elaborated in the same standalone depth as Chow rank and border Chow rank themselves.