Jacobian Criterion: Tests and Applications

• Jacobian criterion is a family of first-order tests using derivative matrices and ideals to determine regularity, invertibility, and smoothness in various mathematical structures.
• It underpins methods ranging from classical algebraic geometry to modern analyses in chemical reaction networks and free polynomial mappings.
• The criterion’s versatility spans arithmetic schemes, singularity theory, and dynamical systems, offering rigorous conditions for invertibility, cohomological smoothness, and multistationarity exclusion.

The Jacobian criterion is not a single theorem but a family of first-order tests in which Jacobian matrices, determinants, minors, Jacobian ideals, or Jacobian algebras control a geometric, algebraic, or dynamical property. In the literature considered here, it appears as a rank criterion for regularity of varieties and arithmetic schemes, a determinant or derivative criterion for invertibility problems surrounding the Jacobian conjecture, a sign criterion for injectivity in chemical reaction networks, a mixed-characteristic nonsingularity criterion involving $p$-derivations, a criterion for cohomological smoothness of Artin $v$-stacks and for variation of hyperplane sections, and a set of ideal-theoretic and syzygetic criteria in singularity theory and blowup algebra theory (Feng et al., 9 Oct 2025, Hochster et al., 2021, Pascoe, 2013, Joshi et al., 2011, Hamann, 2022, Ilardi et al., 21 Jun 2026, Niu et al., 2014, Burity et al., 2023).

1. Regularity, closed points, and mixed characteristic

In its classical algebraic-geometric form, the Jacobian criterion tests regularity by the rank of a matrix of first derivatives. For a variety

$X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$

over a field $K$, and a rational point $x$, the Jacobian matrix is

$$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$

and $x$ is regular if and only if

$\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$

The same paper extends this from rational points to arbitrary closed points by replacing the ambient coordinates with generators $g_1,\ldots,g_n$ of the maximal ideal and using

$$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$

again with the condition

$v$0

It also gives an arithmetic variant for $v$1 over a DVR $v$2 with uniformizer $v$3, where the relevant matrix is the augmented Jacobian

$v$4

and regularity is equivalent to

$v$5

The same source further states that regularity is preserved under finite separable base change over fields, under unramified arithmetic base change, and in the ramified arithmetic case precisely when the special fiber is regular (Feng et al., 9 Oct 2025).

A mixed-characteristic refinement replaces the classical derivative-only matrix by a “mixed Jacobian” incorporating a $v$6-derivation. For an essentially affine algebra over a complete local algebra over a mixed characteristic DVR, nonsingularity at primes containing $v$7 is controlled by the $v$8 minors of a matrix whose rows are indexed by generators $v$9 and whose columns include a $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$0-column with entries $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$1, the ordinary partial derivatives, and additional columns attached to a $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$2-base of the residue field. The criterion states that $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$3 is nonsingular if and only if the prime $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$4 does not contain those minors. To linearize the $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$5-derivational component, the paper introduces perivations and a universal perivation module $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$6, which plays the role of Kähler differentials in this setting. It also proves a freeness criterion: $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$7 This places the mixed-characteristic Jacobian criterion alongside the classical one, but with genuinely new arithmetic data in the matrix (Hochster et al., 2021).

2. Invertibility and the Jacobian conjecture

In the polynomial automorphism problem, the Jacobian criterion is the condition that a polynomial map has everywhere nonzero constant Jacobian determinant. In the commutative setting, for a polynomial map $X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$8, the criterion is

$X = K[t_1,\ldots,t_n]/(f_1,\ldots,f_r)$9

and the Jacobian conjecture asks whether this implies invertibility with polynomial inverse (Pascoe, 2013). The same theme appears in the form

$K$0

for polynomial maps $K$1, with special emphasis on cubic reductions such as Druzkowski mappings (Yan, 2012).

For the two-dimensional conjecture, one paper states that if $K$2 is a pair of Jacobian polynomials, then

$K$3

where $K$4 is the intersection number of $K$5 in the affine plane, $K$6 is the number of branch at point at infinity, and $K$7 is the geometric genus of the affine curve defined by $K$8. The stated consequence is that every Jacobian polynomial defines a smooth rational curve with one point at infinity, and that this is sufficient to fix the Jacobian conjecture in two dimension by the Abhyankar theorem or the Abhyankar-Moh-Suzuki theorem (Joe, 2013).

A recent perturbative formulation studies maps

$K$9

under the Jacobian hypothesis

$x$0

which is presented as equivalent to $x$1. In that framework the inverse satisfies

$x$2

is expanded as a sum of partially ordered connected trees, and is controlled by the derivative identity

$x$3

The paper further states

$x$4

and, using the Jacobian hypothesis, obtains the degree bound

$x$5

This formulation treats the criterion not merely as a determinant condition but as a cancellation mechanism inside a combinatorial expansion of the inverse map (Magnen, 2023).

A recurring distinction in this literature is between local and global force. The determinant condition is the natural first-order hypothesis, but in the classical commutative case it does not by itself settle global polynomial invertibility; by contrast, other settings discussed below turn the same first-order datum into a complete global criterion.

3. Effective reductions and special classes of polynomial maps

One major line of work seeks effective criteria in restricted classes. For Druzkowski mappings

$x$6

if

$x$7

then

$x$8

where $x$9 is the number of diagonal elements of $$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$0 equal to zero. In the special case $$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$1, this gives

$$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$2

and the same paper states that the Jacobian Conjecture is true for such Druzkowski mappings in dimension $$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$3 (Yan, 2012).

A related approach introduces Condition C1 and the weaker Condition C2 for Druzkowski maps. For a cubic linear map

$$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$4

one theorem states that $$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$5 is an automorphism if and only if the only solution of

$$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$6

is $$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$7. The paper further states that, for Druzkowski matrices, Condition C2 is equivalent to the Jacobian conjecture; that C1 holds for a generic matrix of any given rank; that C1 is invariant under diagonal conjugation; and that C2 is invariant under cubic similarity. It also proposes a low-dimensional strategy, including dimension $$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$8, based on these reformulations (Truong, 2015).

In dimension two, a different reduction replaces direct inverse construction by commutative-algebra conditions on an intermediate ring. For a morphism $$J_x=\left(\frac{\partial f_i}{\partial t_j}(x)\right),$$9 with invertible Jacobian, writing $x$0 and $x$1, the paper proves that $x$2 is invertible if any one of the following equivalent conditions holds:

1. $x$3 is normal;
2. $x$4 is flat over $x$5;
3. $x$6 is separable over $x$7. The result is framed as a special-case resolution of the two-variable conjecture by checking ring-theoretic properties of $x$8 (Moskowicz, 2015).

A further effective criterion recasts the problem in Picard-Vessiot terms. For a polynomial map $x$9 with constant nonzero Jacobian determinant, let

$\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$0

Then $\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$1 is a polynomial automorphism if and only if $\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$2 is a fundamental solution matrix for an integrable linear partial differential system

$\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$3

with $\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$4 over $\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$5. The same paper gives the explicit bound

$\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$6

for the number of wronskians that need to be checked, and emphasizes that the method is specially efficient with cubic homogeneous mappings introduced and studied in fundamental papers by H. Bass, E. Connell, D. Wright and L. Druzkowski (Adamus et al., 2015).

4. Free, stacky, and moduli-theoretic variants

In free analysis, the Jacobian criterion becomes global. For a free map $\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$7, the derivative is defined by the block identity

$\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$8

The free inverse function theorem states that the following are equivalent: $\operatorname{rank}(J_x)=n-\dim \mathcal{O}_{X,x}.$9 is nonsingular for every $g_1,\ldots,g_n$0; $g_1,\ldots,g_n$1 is injective; and $g_1,\ldots,g_n$2 exists and is a free map. For free polynomial maps this becomes an equivalence between everywhere-nonsingular derivative, injectivity, bijectivity, and the existence of polynomial inverses on each matrix size. The paper also gives an analogous criterion for commutative polynomials evaluated on tuples of commuting matrices, where global nonsingularity on commuting tuples is equivalent to injectivity, bijectivity, and polynomial invertibility (Pascoe, 2013).

A different extension concerns variation of hyperplane sections of hypersurfaces through the Milnor algebra

$g_1,\ldots,g_n$3

For a hypersurface $g_1,\ldots,g_n$4, the hyperplane-section map $g_1,\ldots,g_n$5 is studied infinitesimally by

$g_1,\ldots,g_n$6

The criterion stated is that $g_1,\ldots,g_n$7 is generically finite onto its image if and only if $g_1,\ldots,g_n$8 is injective for a general hyperplane. In the smooth case this recovers the Lefschetz criterion via injectivity of

$g_1,\ldots,g_n$9

for a general linear form $$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$0. In the singular case a new obstruction appears: a linear Jacobian syzygy, equivalently, for non-cones, a positive-dimensional projective automorphism group. Once that obstruction is excluded, maximal infinitesimal variation is governed by the same Lefschetz map $$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$1 (Ilardi et al., 21 Jun 2026).

On the Fargues–Fontaine curve, the Jacobian criterion is reformulated for section spaces of stack quotients $$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$2. Let $$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$3 denote the locus of sections $$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$4 for which the pullback tangent complex is represented by a two-term complex

$$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$5

with $$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$6 of strictly positive slopes. The criterion states that this locus is cohomologically smooth, and that its local $$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$7-dimension at a geometric point is

$$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$8

or equivalently

$$J_x=\left(\frac{\partial f_i}{\partial g_j}(x)\right),$$9

Here the Jacobian criterion is expressed through tangent complexes and positivity of slopes rather than through ordinary determinants (Hamann, 2022).

5. Jacobian ideals, schemes, syzygies, and multiplier ideals

When the Jacobian data are organized ideal-theoretically, the criterion measures singularities rather than invertibility. For a reduced singular projective plane curve $v$00, the Jacobian scheme is the zero-dimensional scheme defined by

$v$01

and its degree is the global Tjurina number

$v$02

If $v$03 is the minimal degree of a Jacobian syzygy, then the du Plessis–Wall bounds are

$v$04

For conic-line arrangements with $v$05, the paper gives a complete geometric characterization of those attaining the upper bound, and also characterizes those attaining the lower bound among all curves with a linear Jacobian syzygy. It further identifies when the Jacobian scheme is an eigenscheme of a ternary tensor, relating the criterion to polar maps and determinantal Jacobian ideals (Beorchia et al., 2023).

Mather–Jacobian multiplier ideals provide another Jacobian-based singularity invariant. Given a resolution $v$06 factoring through the Nash blow-up, a nonzero ideal $v$07, and $v$08,

$v$09

The paper proves the comparison

$v$10

with equality when $v$11 is a locally complete intersection. On algebraic curves, where the definition extends to any characteristic, it shows that

$v$12

It also proves the conductor criterion

$v$13

and records the inclusions

$v$14

for curves (Niu et al., 2014).

Blowup algebra theory studies the Jacobian ideal

$v$15

through its Rees algebra $v$16, special fiber $v$17, and analytic spread

$v$18

For several new families of homogeneous free divisors, the Rees algebra and special fiber are shown to be Cohen-Macaulay. More generally, for a reduced homogeneous polynomial $v$19 with linearly independent partials, the paper characterizes maximal analytic spread $v$20 by several equivalent conditions, including full rank of the Hessian map, $v$21-linear independence of the gradients $v$22, and the asymptotic depth condition

$v$23

for some large $v$24. It also gives an ideal-theoretic homological criterion for homaloidal divisors, כלומר hypersurfaces whose polar maps are birational (Burity et al., 2023).

6. Chemical, real-dynamical, and billiard interpretations

In chemical reaction network theory, the Jacobian criterion is a sign condition on the determinant expansion of a parametrized Jacobian matrix. A network passes the Jacobian Criterion if all terms in that expansion have the same sign, and passing the criterion rules out multiple positive steady states. The determinant can be expanded over $v$25-square subnetworks, and for a square subnetwork $v$26,

$v$27

with orientation

$v$28

A principal simplification is that one need only examine certain embedded square networks rather than all $v$29-square subnetworks. The paper also gives a new sufficient condition: if all species of a CFSTR have total molecularity at most two, then the network passes the Jacobian Criterion and therefore cannot admit multiple steady states (Joshi et al., 2011).

In the real Jacobian problem, one paper converts injectivity into a dynamical criterion at infinity. For a polynomial map $v$30 with $v$31, it associates the Hamiltonian

$v$32

and proves that global injectivity is equivalent to each of the following conditions for the Bendixson compactified system: the origin is a center; the origin is monodromic; the origin has no hyperbolic sectors; and the system has a $v$33 first integral with an isolated minimum at the origin. The same paper introduces the criterion function

$v$34

and states that $v$35 is injective if and only if

$v$36

In $v$37 dimensions, it gives the analogous condition

$v$38

for polynomial maps with nowhere vanishing Jacobian determinant (Tian et al., 2020).

A further reinterpretation arises in static billiards. In canonical variables, area preservation corresponds to $v$39, but in noncanonical angular coordinates $v$40 the Jacobian determinant is generally not identically unity. The paper gives the explicit formula

$v$41

and interprets the regions $v$42 and $v$43 as local phase-space expansion and contraction. The curves $v$44 act as deformation boundaries, intersect unstable periodic points, and correlate with invariant manifolds; for period-two orbits the composed map restores exact unit determinant,

$v$45

This use of the Jacobian criterion is not an injectivity test but a geometric diagnostic of deformation in conservative dynamics (Fonseca et al., 8 Mar 2026).

Across these contexts, the common mathematical motif is that Jacobian data encode first-order behavior, while the property extracted from that data depends sharply on the ambient category. In some settings the criterion remains local and detects regularity or nonsingularity; in others it becomes global and characterizes invertibility, injectivity, cohomological smoothness, variation in moduli, or exclusion of multistationarity. The resulting phrase “Jacobian criterion” is therefore best understood as a family resemblance: a technical idiom in which derivatives, minors, syzygies, and Jacobian algebras serve as the primary invariants for first-order control.