Teissier's Jacobian Newton Polygon

• Teissier’s Jacobian Newton polygon is a convex geometric invariant derived from the discriminant of a holomorphic mapping, encoding polar branch multiplicities, intersection numbers, and Łojasiewicz exponents.
• It is constructed using ideal-theoretic methods and Minkowski sums, ensuring invariance under equisingularity and serving as a key tool in singularity analysis.
• Recent extensions include alternating versions and higher-dimensional applications that enable precise computation of invariants like sectional Milnor numbers and optimal gradient exponents.

Searching arXiv for the cited papers and closely related work on Jacobian Newton polygons. Teissier’s Jacobian Newton polygon is the Newton polygon of the discriminant of a holomorphic mapping associated with a singularity, classically the map $(f,z_1)$ for an isolated hypersurface singularity and, in the plane two-function setting, the map $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$. It packages polar data into a convex-geometric object in $\mathbb{R}^2$: multiplicities of polar branches, intersection numbers, Jacobian quotients, sectional Milnor numbers, and Łojasiewicz exponents. In Teissier’s formulation it is also the Newton polygon of the pair of ideals $(J(f),\mathfrak m)$, while later work proves equisingularity invariance for arbitrary plane pairs, constructs families of such polygons from approximate roots of a branch, and introduces an alternating version with explicit resolution and Newton-diagram formulas (Teissier, 2012, Gwozdziewicz, 2011, Barroso et al., 2011, Sigurðsson, 7 Sep 2025).

1. Definition and basic constructions

In two variables, for a power series

$$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$

its Newton diagram is

$$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$

with $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$. In dimension $2$, this is exactly the usual Newton polygon. For a germ

$$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$

the Jacobian determinant is

$$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$

Its zero locus is the critical locus of the map. The discriminant curve is the direct image of this critical set by $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$0; if $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$1 is an equation of that image, then the Jacobian Newton diagram of $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$2 is $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$3. In the terminology emphasized by Gwoździewicz, this is exactly Teissier’s Jacobian Newton polygon in the plane case (Gwozdziewicz, 2011).

For an isolated hypersurface singularity

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$4

Teissier chooses a generic linear form $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$5 and studies the map

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$6

The polar curve is defined by

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$7

with branches $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$8, multiplicities $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$9, and intersection data

$\mathbb{R}^2$0

The discriminant of $\mathbb{R}^2$1 has Newton polygon

$\mathbb{R}^2$2

where each term denotes an elementary Newton polygon. Teissier also identifies this polygon with the Newton polygon of the pair of ideals $\mathbb{R}^2$3: $\mathbb{R}^2$4 This shifts the construction from explicit equations of discriminants to an ideal-theoretic invariant of the embedded germ $\mathbb{R}^2$5 (Teissier, 2012).

2. Polar data, Minkowski structure, and encoded invariants

A basic structural feature is additivity under Minkowski sum. If

$\mathbb{R}^2$6

is the factorization of the Jacobian determinant into irreducible factors, then the plane Jacobian Newton diagram decomposes as

$\mathbb{R}^2$7

where $\mathbb{R}^2$8 is an elementary Newton diagram and $\mathbb{R}^2$9 is the intersection multiplicity at the origin. The inclination of $(J(f),\mathfrak m)$0 is $(J(f),\mathfrak m)$1, and the set of inclinations is independent of the chosen decomposition. These inclinations are exactly the Jacobian quotients, or Hironaka numbers,

$(J(f),\mathfrak m)$2

attached to the irreducible factors of $(J(f),\mathfrak m)$3. In the polar case $(J(f),\mathfrak m)$4 with $(J(f),\mathfrak m)$5 smooth and generic, they are the polar quotients of Teissier’s theory (Gwozdziewicz, 2011).

In Teissier’s hypersurface setting, the same polygon controls several numerical invariants. For generic sections $(J(f),\mathfrak m)$6, the associated Jacobian polygons satisfy

$(J(f),\mathfrak m)$7

with $(J(f),\mathfrak m)$8. Thus length and height encode the sequence of sectional Milnor numbers. Teissier also proves that every slope $(J(f),\mathfrak m)$9 of the Jacobian polygon satisfies

$$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$0

and that the Łojasiewicz exponent is the degree minus $$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$1: $$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$2 In the 1978 notes, the slopes further control the optimal exponents in gradient inequalities,

$$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$3

with

$$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$4

and also the jet order required for topological stability under perturbations $$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$5 (Teissier, 2012).

3. Equisingularity invariance for plane pairs

A central plane result is Gwoździewicz’s theorem: if

$$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$6

then the Jacobian Newton diagram of $$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$7 depends only on the equisingularity class of the pair of curves $$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$8 and $$h(x,y)=\sum_{i,j} c_{ij}x^i y^j \in \mathbb{C}\{x,y\},$$9. Here equisingularity is the topological-embedded notion for pairs of plane curve germs, equivalently encoded by the embedded resolution graph together with the multiplicity data of $$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$0 and $$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$1 along exceptional components and strict transforms. The theorem says that Teissier’s Jacobian Newton polygon is constant on equisingularity classes of pairs, not merely on analytic equivalence classes (Gwozdziewicz, 2011).

The proof combines two ingredients. First, for all $$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$2 except finitely many, the equisingularity class of the generic member of the pencil

$$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$3

depends only on the equisingularity class of the pair $$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$4. This is read from a minimal good resolution $$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$5 of $$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$6, through the orders

$$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$7

and the local models

$$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$8

at intersections of components with $$\Delta_h=\operatorname{Conv}\Bigl(\bigcup_{\{(i,j):\,c_{ij}\neq 0\}}\bigl((i,j)+\mathbb{R}_+^2\bigr)\Bigr),$$9 and $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$0. Second, for coprime positive integers $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$1, the support function of the Jacobian Newton diagram satisfies

$\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$2

for generic $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$3. Since the Milnor number of the generic curve $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$4 is determined by its equisingularity class, the support function depends only on the pair’s equisingularity class; because the support function determines the Newton diagram, the entire polygon is invariant (Gwozdziewicz, 2011).

The scope of this statement is specific. It applies to analytic maps $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$5 with isolated fiber over $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$6, and it does not claim that the Jacobian Newton polygon is a complete invariant of equisingularity for pairs. The result is constancy on equisingularity classes, not classification of such classes (Gwozdziewicz, 2011).

4. Plane branches, approximate roots, and completeness phenomena

For an irreducible plane branch $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$7, García Barroso and Gwoździewicz define the $\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$8-th approximate Jacobian Newton diagram as the Jacobian Newton diagram of the morphism

$\mathbb{R}_+=\{x\in\mathbb{R}:x\ge 0\}$9

where $2$0 is the $2$1-th characteristic approximate root of $2$2. If $2$3 is an irreducible Weierstrass polynomial with Puiseux characteristic $2$4, and

$2$5

then $2$6 is the $2$7-th approximate root of $2$8. Each $2$9 is irreducible with Puiseux characteristic

$$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$0

degree $$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$1, and contact

$$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$2

Thus the sequence of approximate roots gives a canonical family of morphisms to which Teissier’s construction can be applied (Barroso et al., 2011).

The paper’s abstract states that the set of all approximate Jacobian Newton diagrams is a complete topological invariant of a branch. In the body of the paper, Corollary 2 states that the family of approximate Jacobian Newton diagrams of a branch depends only on its topological type, and the reconstruction of the semigroup from the full family yields the converse direction. The case $$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$3 recovers the classical Teissier–Merle–Ephraim setting of a smooth transverse branch $$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$4, while higher $$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$5 refine the polar data through the intrinsic sequence of approximate roots. The critical curve

$$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$6

admits a factorization

$$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$7

and the canonical decomposition of $$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$8 is expressed through the intersection numbers with these factors. The semigroup generators $$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0), \qquad (f,g)^{-1}(0,0)=\{(0,0)\},$$9 can be recovered from the family $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$0, so the family determines the Puiseux characteristic and hence the embedded topological type (Barroso et al., 2011).

The examples in that paper also delimit the invariant’s completeness. The entire family is complete, but removing $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$1 can destroy completeness: different branches may share the “tail” $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$2. A further example shows that the contacts between branches of the Jacobian and the original branch can depend on analytic structure rather than only on the Puiseux characteristic, which explains why the discriminant Newton diagram, rather than naive factor-contact data alone, is the stable object (Barroso et al., 2011).

5. Equisingularity, mixed multiplicities, and Thom–Sebastiani structure

In Teissier’s 1978 notes, the Jacobian Newton polygon is presented as a refinement of the Milnor number and as an invariant naturally attached to equisingularity questions. For a family of hypersurfaces with isolated singularities and constant total topological type—the collection of embedded topological types of general $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$3-plane sections for all $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$4—Teissier proves that the Jacobian Newton polygon is constant in the family. He also remarks that in families the polygon is upper semicontinuous “in the sense of polygons being above,” and asks whether constancy of $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$5 already forces constancy of the maximal slope (Teissier, 2012).

The ideal-theoretic formulation makes the polygon part of the theory of mixed multiplicities. For two $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$6-primary ideals $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$7 in a Cohen–Macaulay local analytic algebra, Teissier defines a Newton polygon $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$8 by valuations along exceptional divisors of the normalized blowing-up of $$J(f,g)=\frac{\partial f}{\partial u}\frac{\partial g}{\partial v}-\frac{\partial f}{\partial v}\frac{\partial g}{\partial u}.$$9. When $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$00 is generated by $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$01 general elements, one obtains a reduced curve $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$02 with components $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$03, and

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$04

Its height and length are mixed multiplicities: $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$05 Specializing to $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$06 gives the Jacobian Newton polygon of a hypersurface. Teissier also proves invariance under integral closure: $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$07 This places the polygon alongside mixed multiplicities, polar multiplicities, and integral closure as part of a common valuation-theoretic package (Teissier, 2012).

The appendix of the same work equips Newton polygons with a semiring structure. Addition is Minkowski-sum-like, every polygon admits a canonical decomposition into elementary polygons, and there is a product $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$08 defined first on elementary polygons and extended distributively. For ideals in completed tensor products, Teissier proves

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$09

where

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$10

For a Thom–Sebastiani sum

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$11

this yields

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$12

A plausible implication is that the Jacobian Newton polygon is not only an equisingularity invariant but also a functorial object with respect to standard singularity operations, in a way that mirrors the behavior of mixed multiplicities and of other Thom–Sebastiani phenomena (Teissier, 2012).

6. Alternating Jacobian polygons and recent higher-dimensional extensions

A recent extension introduces an alternating version of the Jacobian polygon for isolated complex hypersurface singularities. If

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$13

denotes the Jacobian polygon of a generic $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$14-dimensional section, the alternating Jacobian polygon is

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$15

It satisfies

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$16

and

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$17

Moreover, $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$18 iff $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$19 is odd and $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$20 is Morse. The construction is linked to a Morse function on the Milnor fiber whose critical points are intersections with polar curves in generic linear sections, yielding a Morse–Smale complex filtered by Teissier’s vanishing rates $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$21. The filtration is compatible with the differential because flow lines can only go from smaller to larger or equal vanishing rate (Sigurðsson, 7 Sep 2025).

The main payoff is computational. For an embedded resolution $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$22 with exceptional components $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$23, multiplicities

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$24

and strict transforms $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$25, one has

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$26

and

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$27

For Newton nondegenerate $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$28, the alternating polygon admits a pure volume formula over coordinate facets $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$29 of the Newton diagram: $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$30 By contrast, the non-alternating polygon involves mixed volumes. The degree of the alternating polygon equals the degree of the ordinary Jacobian polygon, so—except when $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$31 is odd and $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$32 is Morse—

$(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$33

In the Newton nondegenerate case this yields a global formula for the Łojasiewicz exponent in terms of Newton numbers of subdiagrams, and the paper provides a counterexample to the Brzostowski–Krasiński–Oleksik conjecture in dimension $(f,g):(\mathbb{C}^2,0)\to(\mathbb{C}^2,0)$34, replacing a facet-by-facet criterion with a global threshold construction over the Newton diagram (Sigurðsson, 7 Sep 2025).

These higher-dimensional developments do not replace Teissier’s original polygon; rather, they recast it. The ordinary Jacobian polygon remains the basic polar invariant, while the alternating polygon isolates the part of the resolution data that is computable by pure face volumes in the Newton nondegenerate case. This suggests a contemporary reading of Teissier’s program: the Jacobian Newton polygon is simultaneously a discriminant invariant, a mixed-multiplicity invariant, a Morse-theoretic invariant, and a Newton-polyhedral invariant, with each viewpoint emphasizing a different aspect of equisingularity and singularity-theoretic stability (Sigurðsson, 7 Sep 2025).