Bivariate Vandermonde Determinants

• Bivariate Vandermonde determinants are matrix constructions built from bivariate monomials or polynomial families that exhibit antisymmetry and factorization properties similar to classic univariate Vandermonde matrices.
• They arise in diverse contexts—from providing an explicit Catalan-indexed basis for alternating diagonal coinvariants to enabling separation of variables on tensor-product grids and analysis via homogeneous evaluation matrices.
• Elegant separation formulas and tableau-indexed identities highlight their versatility, linking combinatorial Dyck path statistics with applications in interpolation, projective geometry, and symmetric-function expansions.

Searching arXiv for the specified paper and closely related work on bivariate and multivariable Vandermonde determinants. arXiv search query: "bivariate Vandermonde determinants diagonal coinvariant ring Dyck paths Jiang" Bivariate Vandermonde determinants are determinant constructions in which the classical univariate Vandermonde matrix is replaced by a matrix built from bivariate monomials, bivariate polynomial families, or products of linear forms evaluated at pairs of variables or points. In the recent literature, the term encompasses several non-equivalent but structurally related objects: determinants of the form $\Delta_X(x,y)=\det[x_i^{\alpha_j}y_i^{\beta_j}]$, grid-Vandermonde matrices with rows $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$, evaluation matrices $[p(a_r,b_s)]$ for homogeneous $p(x,y)$, and projective Veronese-block determinants attached to linear forms. These constructions share the characteristic feature that antisymmetry, collision loci, and factorization phenomena are controlled by one-variable Vandermonde factors, but they arise in different mathematical settings, including diagonal coinvariants, interpolation on tensor-product grids, symmetric-function expansions, and projective intersection theory (Jiang, 17 Aug 2025).

1. Definitions and principal matrix models

One standard definition fixes $n$ and a sequence of $n$ pairs of nonnegative integers

$$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$

and sets

$$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$

Equivalently, if $X$ is viewed as an $n\times 2$ matrix of exponents, then $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$0 is the determinant of the $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$1 matrix whose $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$2-entry is $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$3 (Jiang, 17 Aug 2025). This is the form used in the study of alternating diagonal coinvariants.

A second model arises from tensor-product grids. If $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$4, $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$5, and $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$6, $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$7, then the Hadamard product $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$8 defines bivariate polynomials $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$9, and the associated Vandermonde matrix on $[p(a_r,b_s)]$0 has determinant

$[p(a_r,b_s)]$1

This is the framework in which variables separate on almost-square tensor-product grids (Marchi et al., 2013).

A third formulation starts from a homogeneous polynomial

$[p(a_r,b_s)]$2

and vectors $[p(a_r,b_s)]$3, $[p(a_r,b_s)]$4. The evaluation matrix

$[p(a_r,b_s)]$5

admits the factorization

$[p(a_r,b_s)]$6

where $[p(a_r,b_s)]$7, $[p(a_r,b_s)]$8, and $[p(a_r,b_s)]$9 (Jitman et al., 24 Jan 2026).

A fourth model is the grid-Vandermonde matrix

$p(x,y)$0

whose $p(x,y)$1th row is

$p(x,y)$2

This is the bivariate specialization of multivariable Vandermonde determinants obtained from amalgams of matrices (Brown, 22 Apr 2026).

Framework	Determinant / matrix	Structural feature
Exponent-pair determinant	$p(x,y)$3	Alternating basis elements in $p(x,y)$4
Separated-variable grid	$p(x,y)$5	Product factorization into univariate determinants
Homogeneous evaluation matrix	$p(x,y)$6	Rank bound and closed square-case formula
Grid-Vandermonde	$p(x,y)$7	Separation as a sum of factorizing terms

The coexistence of these models shows that the phrase “bivariate Vandermonde determinant” does not denote a single canonical object. This suggests that the subject is best understood as a family of determinant identities organized by the choice of bivariate basis and by the geometry of the evaluation set.

2. Alternating diagonal coinvariants and Dyck-path indexing

In the diagonal-coinvariant setting, Jiang gives an explicit basis for the alternating part $p(x,y)$8 of the diagonal coinvariant ring $p(x,y)$9 in terms of bivariate Vandermonde determinants, answering a question of Stump (Jiang, 17 Aug 2025). The indexing set is the set

$n$0

of ordinary Dyck paths of semilength $n$1.

For each $n$2, two integer sequences of length $n$3 are defined. The first is the usual area-sequence

$n$4

The second is the dinv-sequence

$n$5

so that

$n$6

From these statistics one forms

$n$7

The main theorem states that

$n$8

is a $n$9-basis of the alternating component $n$0 of $n$1 (Jiang, 17 Aug 2025). Concretely,

$n$2

Each such determinant is manifestly alternating under the simultaneous permutation of the index set $n$3 in $n$4 and in $n$5, so it transforms by the sign character under the diagonal $n$6-action.

The proof combines several structural facts. No two pairs $n$7 coincide for a single $n$8, so each determinant is nonzero. If $n$9, then $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$0 and $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$1 are distinct as multisets of pairs, so the determinants are distinct. Finally, $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$2 is the $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$3th Catalan number $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$4, which equals $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$5. These facts together show that the Catalan-many determinants are linearly independent and span exactly $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$6 (Jiang, 17 Aug 2025).

The significance of this construction is twofold. First, it gives an explicit $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$7-alternating basis formed by bivariate Vandermonde-type determinants. Second, it provides a direct combinatorial description of the alternating part of the diagonal coinvariant ring in terms of Dyck-path statistics. In this setting, bivariate Vandermonde determinants are not merely interpolation devices; they are canonical basis vectors in a representation-theoretic object.

3. Separation of variables on tensor-product grids

A distinct line of work studies bivariate Vandermonde determinants on tensor-product grids whose variables separate. Let $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$8, $$X=((\alpha_1,\beta_1),(\alpha_2,\beta_2),\dots,(\alpha_n,\beta_n)),$$9, with $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$0 or $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$1, and suppose $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$2 and $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$3 satisfy the block-semiseparable structure described in equation (1) of the paper, with $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$4 and $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$5 monic of degree $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$6 for $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$7. Then

$$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$8

where $$\Delta_X(x,y)=\det\bigl[x_i^{\alpha_j}y_i^{\beta_j}\bigr]_{1\le i,j\le n}.$$9 is the univariate Vandermonde determinant of the $X$0th column of $X$1 over $X$2, and similarly for $X$3 (Marchi et al., 2013).

The proof proceeds by induction. The base case $X$4 is direct. For general $X$5 and $X$6, the $X$7 Vandermonde matrix is written in block form, a univariate factor $X$8 is extracted, and the Schur-complement formula is applied to remove the first row-block and first column-block. This introduces a factor $X$9 and reduces the size from $n\times 2$0 to $n\times 2$1. After interchanging the roles of $n\times 2$2 and $n\times 2$3, the induction hypothesis applies. A key one-variable identity is

$n\times 2$4

where

$n\times 2$5

This reduction shows how two-variable structure can collapse to one-variable determinants under strong separation hypotheses (Marchi et al., 2013).

The principal application is to Padua and Padua-like points. For even $n\times 2$6, the Padua-like set is written as $n\times 2$7, and for the classical Padua choice $n\times 2$8 and $n\times 2$9 one obtains

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$00

with

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$01

The second factor is exactly of the separated-variable type, so Proposition 3.1 applies and yields a complete factorization of the Padua Vandermonde determinant into one-dimensional determinants in $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$02 and $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$03 (Marchi et al., 2013). This proves in full generality an explicit factorization conjectured in Bos–Vianello (2009).

A frequent misconception is that every bivariate Vandermonde determinant should factor as a single product of univariate Vandermondes. The separated-variable theory shows that such factorization is available under a specific block-semiseparable hypothesis on the polynomial basis and on almost-square tensor-product grids; it is not presented as a universal phenomenon.

4. Homogeneous polynomial evaluation matrices

For a homogeneous polynomial

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$04

the evaluation matrix $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$05 provides a particularly transparent bridge between bivariate and univariate Vandermonde determinants. Because

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$06

one has

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$07

Hence if $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$08, then $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$09 is an $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$10 matrix of rank at most $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$11, and therefore

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$12

(Jitman et al., 24 Jan 2026).

The borderline case $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$13 is square, and the determinant becomes explicit:

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$14

Here $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$15 is the usual Vandermonde in ascending powers, while $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$16 differs from the standard form only by reversing columns, which contributes the sign $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$17 (Jitman et al., 24 Jan 2026).

Several classical determinants appear as special cases. For $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$18, one recovers the determinant formula with coefficient product $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$19, and vanishing for $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$20. For

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$21

one has $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$22 and

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$23

when $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$24, so the evaluation matrix agrees with the classical quotient-Vandermonde matrix. For

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$25

the sign from $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$26 cancels the sign from $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$27, and the determinant is exactly the product of the two ordinary Vandermondes (Jitman et al., 24 Jan 2026).

When $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$28, the factorization is rectangular, so the determinant is expanded by Cauchy–Binet over $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$29-element subsets $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$30. Each minor is a generalized Vandermonde determinant, and standard symmetric-function identities rewrite these minors using complete homogeneous symmetric polynomials $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$31. This yields a closed expansion

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$32

In particular, if the support of the $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$33 has size $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$34, then $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$35 (Jitman et al., 24 Jan 2026).

The same paper also treats the sum-form $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$36, obtaining an explicit determinant formula and an equivariance law under linear changes of variables in $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$37, and derives a non-vanishing bound over finite fields via the Schwartz–Zippel lemma (Jitman et al., 24 Jan 2026). In this framework, the bivariate determinant is controlled entirely by the one-variable Vandermonde determinants in the evaluation vectors and by the coefficient data of $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$38.

5. Multivariable separation formulae and the bivariate grid-Vandermonde

Brown derives two general separation formulae for multivariable Vandermonde determinants by expressing the full matrix as an amalgam, or rowwise Kronecker product, of smaller Vandermonde matrices. In the bivariate case, with $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$39, $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$40, the determinant is attached to

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$41

an $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$42 matrix whose $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$43th row is

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$44

The first formula expresses $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$45 as a sum over pairs of standard tableaux

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$46

with integer coefficients $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$47, and each term is a product of determinants in fewer variables:

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$48

The second formula is a fully symmetric $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$49-sum with sign $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$50 and hook-length normalization

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$51

Both identities decompose the bivariate determinant into completely factorising one-variable Vandermonde factors (Brown, 22 Apr 2026).

The $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$52 case is especially explicit. Here

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$53

and the first separation formula reduces to the two-term identity

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$54

The symmetric form is

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$55

Each term manifestly factors as a product of four one-variable Vandermonde factors in the $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$56’s and $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$57’s respectively (Brown, 22 Apr 2026).

This result is structurally different from the separated-variable product formula of the tensor-grid setting. Here the determinant is not represented as one product but as a signed sum of completely factorising terms. That distinction is important: it shows that “separation” can mean exact product factorization in some regimes and tableau-indexed additive decomposition in others.

6. Projective identities, Vandermonde-like analogues, and open directions

A projective version of the subject appears in the bivariate specialization of the higher-dimensional Vandermonde determinant identity. In projective $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$58-space with homogeneous coordinates $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$59, let

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$60

and form the Veronese-block matrix $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$61 whose rows are the coefficient vectors of all products

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$62

over all $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$63 with $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$64. Then

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$65

In particular, $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$66 vanishes exactly when some triple of the lines $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$67 in $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$68 become concurrent (Yaacov, 2014). Here the Vandermonde principle is expressed not through evaluation of monomials at points, but through products of linear forms and a universal concurrency detector.

A different analogue is provided by the CI-matrix, whose entries are elementary symmetric polynomials in all variables except one. If

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$69

then

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$70

Although this matrix is not bivariate, it shows that classical Vandermonde determinants persist under substantial changes of basis: monomials may be replaced by the “omit-one” elementary-symmetric basis without changing the determinant (Ferrante et al., 2019). This is relevant because many bivariate constructions likewise depend more on collision behavior and degree structure than on a unique preferred basis.

Vandermonde-type determinants also appear in probabilistic applications. The $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$71 determinants

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$72

factor as the full six-factor Vandermonde prefactor

$$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$73

times multiple sums with strictly positive integer coefficients, and therefore vanish exactly when two of $$(1,x_i,\dots,x_i^{a-1})\otimes(1,y_i,\dots,y_i^{b-1})$$74 coincide (Turan et al., 2018). The same work states that finding a fully explicit “bivariate Vandermonde” formula in general remains an open problem.

Taken together, these results delineate the modern scope of the subject. In diagonal coinvariants, bivariate Vandermonde determinants furnish an explicit Catalan basis (Jiang, 17 Aug 2025). On almost-square tensor-product grids, they factor into one-dimensional determinants (Marchi et al., 2013). For homogeneous evaluation matrices, they are governed by rank bounds and closed square-case formulas (Jitman et al., 24 Jan 2026). In the grid-Vandermonde setting, they admit separation identities indexed by tableaux and permutations (Brown, 22 Apr 2026). In projective geometry and related analogues, they detect concurrency and other collision phenomena (Yaacov, 2014). This suggests that the unifying content of the theory lies not in a single determinant formula, but in a common algebra of antisymmetry, collision divisibility, and basis-dependent factorization.