Eggers Decomposition in Plane Curves

• Eggers Decomposition is a combinatorial method that organizes equisingularity data in plane curve germs by structuring branch information into Eggers–Wall trees.
• It encodes key functions—exponent, index, and contact complexity—to recover characteristic exponents, Puiseux denominators, and intersection multiplicities from the tree.
• In positive characteristic, the technique further provides a canonical factorization of polar curves, ensuring arithmetic conditions prevent inseparability issues.

Searching arXiv for the cited Eggers–Wall and related papers to ground the article. First, I’ll look up the 2018 paper on the valuative tree and Eggers–Wall trees. Now I’ll search for the positive-characteristic paper on Eggers decomposition of polar curves. I’ll also check for closely related arXiv records on polar curves and Eggers–Wall trees for context. Eggers decomposition denotes a family of closely related constructions that organize the equisingularity data of plane curve germs by means of an Eggers–Wall tree. In the setting of a reduced curve germ $C$ on a smooth germ $S$ of complex analytic surface with a chosen smooth branch $L$, the construction decomposes the singularity into branchwise characteristic segments glued along their mutual orders of coincidence, and it identifies this finite combinatorial object with a canonical subtree of the valuative tree $\mathbb{P}(\mathcal{V})$ (Barroso et al., 2018). In the setting of a plane curve germ $f(x,y)=0$ over an algebraically closed field of characteristic $p>0$, the same tree-theoretic language supports a distinct but related notion: a canonical factorization of the polar $P_y(f)=\partial f/\partial y$ into factors indexed by the marked points of the Eggers–Wall tree, with existence governed by an arithmetic condition read directly from the tree (Felipe et al., 28 Sep 2025).

1. Conceptual scope

The term refers, first, to a decomposition of singularity data. For a reduced germ $C$ with branches $C_i$, the Eggers–Wall tree $\Theta_L(C)$ organizes the branches as ends, encodes their genealogies by gluing branch segments along initial pieces determined by mutual coincidence orders, and carries three natural functions: the exponent function $S$0, the index function $S$1, and the contact complexity $S$2. In this sense, Eggers decomposition is a combinatorial and metric encoding of characteristic exponents, least common denominators of Puiseux exponents, and mutual contacts or intersection multiplicities (Barroso et al., 2018).

The term also refers, in the positive-characteristic polar setting, to a canonical partition of the irreducible factors of $S$3 according to their attaching points on the Eggers–Wall tree $S$4. This decomposition is defined by two equivalent conditions, denoted $S$5 and $S$6, which prescribe where factors may attach and what their intersection multiplicities with the $S$7-axis must be. The resulting factorization is canonical relative to $S$8 and the chosen direction $S$9 (Felipe et al., 28 Sep 2025).

These two uses are structurally aligned. In both, the tree is not merely a visual summary: it is the organizing object from which contacts, multiplicities, and change-of-observer behavior are recovered. A plausible implication is that “Eggers decomposition” is best understood as a tree-indexed disassembly of singularity data rather than as a single factorization procedure.

2. Eggers–Wall trees and the three natural functions

Let $L$0 be a smooth germ of complex analytic surface with special point $L$1, let $L$2 be a smooth branch through $L$3, and choose local coordinates $L$4 such that $L$5. For a branch $L$6, one considers a monic irreducible Weierstrass polynomial $L$7 of degree $L$8, together with its Newton–Puiseux roots in $L$9. The characteristic exponents of $\mathbb{P}(\mathcal{V})$0 relative to $\mathbb{P}(\mathcal{V})$1 are the valuations $\mathbb{P}(\mathcal{V})$2 for distinct Newton–Puiseux roots $\mathbb{P}(\mathcal{V})$3, or equivalently the exponents in one Puiseux root that strictly increase the common denominator (Barroso et al., 2018).

For each branch $\mathbb{P}(\mathcal{V})$4, the Eggers–Wall segment $\mathbb{P}(\mathcal{V})$5 is a compact oriented segment with an increasing homeomorphism

$\mathbb{P}(\mathcal{V})$6

called the exponent function. Its marked points are the end labeled by $\mathbb{P}(\mathcal{V})$7, the end labeled by $\mathbb{P}(\mathcal{V})$8, and the points whose exponent values are the characteristic exponents of $\mathbb{P}(\mathcal{V})$9. The index function

$f(x,y)=0$0

assigns to a point $f(x,y)=0$1 the index of $f(x,y)=0$2 in the subgroup of $f(x,y)=0$3 generated by $f(x,y)=0$4 and the characteristic exponents $f(x,y)=0$5; equivalently, it is the least common denominator of exponents of a Puiseux root of $f(x,y)=0$6 strictly less than $f(x,y)=0$7.

For distinct branches $f(x,y)=0$8, their order of coincidence is

$f(x,y)=0$9

The Eggers–Wall tree $p>0$0 of a reduced germ $p>0$1 is obtained by gluing the segments $p>0$2 along their initial parts up to exponent $p>0$3. Proposition 3.12 states that $p>0$4, with $p>0$5 and $p>0$6, depends only on $p>0$7, not on the choice of $p>0$8 (Barroso et al., 2018).

The third natural function is the contact complexity

$p>0$9

Along each branch segment, $P_y(f)=\partial f/\partial y$0 is an increasing homeomorphism to $P_y(f)=\partial f/\partial y$1, piecewise affine and concave in $P_y(f)=\partial f/\partial y$2, while $P_y(f)=\partial f/\partial y$3 is continuous, piecewise affine and convex in $P_y(f)=\partial f/\partial y$4. The functions $P_y(f)=\partial f/\partial y$5 glue to a continuous strictly increasing surjection

$P_y(f)=\partial f/\partial y$6

Function	Codomain	Encoded data
$P_y(f)=\partial f/\partial y$7	$P_y(f)=\partial f/\partial y$8	characteristic exponents along branch segments
$P_y(f)=\partial f/\partial y$9	$C$0	least common denominators accumulated along the segment
$C$1	$C$2	contact complexity and normalized intersection data

The key reconstruction formula is the intersection tripod formula. If $C$3 and $C$4 are distinct branches and $C$5 is the center of the tripod generated by $C$6, then

$C$7

Consequently,

$C$8

Thus the Eggers–Wall tree is already sufficient to recover pairwise intersection multiplicities (Barroso et al., 2018).

3. Valuative realization

A semivaluation on $C$9 is a map $C_i$0 satisfying

$C_i$1

with $C_i$2 for $C_i$3 and $C_i$4. The semivaluation space $C_i$5 is compact, and its projectivization $C_i$6 is the Hausdorff quotient of $C_i$7 by positive scaling, after collapsing the three orbits associated to any branch $C_i$8 into one point. Favre–Jonsson’s theorem equips $C_i$9 with the structure of a compact $\Theta_L(C)$0-tree (Barroso et al., 2018).

Relative to the observer $\Theta_L(C)$1, one works on the section $\Theta_L(C)$2 of semivaluations normalized by $\Theta_L(C)$3, whose root is $\Theta_L(C)$4. For $\Theta_L(C)$5 and $\Theta_L(C)$6, one introduces the closed ball

$\Theta_L(C)$7

and defines

$\Theta_L(C)$8

This yields semivaluations in $\Theta_L(C)$9. If $S$00 lies on the segment $S$01 at exponent $S$02, the valuative embedding sends $S$03 to $S$04. The resulting map

$S$05

is an increasing embedding of rooted trees, sending the root $S$06 to $S$07 and each end $S$08 to the normalized intersection semivaluation $S$09 (Barroso et al., 2018).

Under this embedding, the three natural functions on $S$10 become restrictions of natural valuative functions: $S$11 Here $S$12 is the relative log-discrepancy, $S$13 the relative multiplicity, and $S$14 the relative self-interaction. This identification is central. It shows that the combinatorics of the Eggers–Wall tree are not ad hoc; they are the finite-tree traces of ambient valuative invariants on $S$15.

The same paper proves a projective-limit statement. If one fixes the observer branch $S$16 and ranges over all finite sets of branches $S$17, each yielding a reduced curve $S$18, then the images $S$19 form a projective system under the attaching maps induced by subtree inclusion. The natural map

$S$20

is a homeomorphism. Via $S$21, the valuative tree is therefore the projective limit of Eggers–Wall trees (Barroso et al., 2018).

4. Change of observer and inversion

Eggers decomposition depends on a distinguished observer branch, but the dependence is controlled. If $S$22 and $S$23 are smooth branches that are components of $S$24, then $S$25 and $S$26 have the same finite affine tree. What changes are the functions $S$27, $S$28, and $S$29, and their transformation is explicit (Barroso et al., 2018).

The inversion theorem gives the formulas. Let $S$30 be the unit point of $S$31, and let $S$32 be the attaching map of the segment $S$33. Then

$S$34

$S$35

and

$S$36

If $S$37, these formulas simplify to

$S$38

with

$S$39

Within the valuative tree, these formulas arise from change-of-observer identities for relative log-discrepancy, self-interaction, and multiplicity. For observers $S$40, the scaling factor $S$41 rescales the relative functions by

$S$42

and multiplicity transforms piecewise according to the location of the point with respect to the segment $S$43. This recasts the classical Abhyankar–Zariski inversion phenomenon as a change of observer inside $S$44. The significance is conceptual: different Eggers–Wall trees are not competing models, but observer-dependent coordinate expressions on a single ambient $S$45-tree (Barroso et al., 2018).

5. Polar curves in positive characteristic

Over an algebraically closed field $S$46 of characteristic $S$47, let $S$48 satisfy $S$49 and $S$50, and write the reduced factorization $S$51. The polar associated to $S$52 is

$S$53

In positive characteristic, Puiseux expansions may fail to exist or be fewer, derivatives can vanish modulo $S$54, and separability along branches can break down. The 2025 paper replaces Puiseux-based methods by an Eggers–Wall tree built from semigroups and key polynomials, and uses it to formulate a necessary and sufficient criterion for the polar to admit Eggers decomposition (Felipe et al., 28 Sep 2025).

For an irreducible branch $S$55, the tree $S$56 is a compact segment rooted at $S$57, equipped with a strictly increasing contact complexity

$S$58

marked points determined by key polynomials, and a piecewise constant index function $S$59. For reducible $S$60, the trees $S$61 are glued along initial segments according to the strong triangle inequality for the logarithmic distance

$S$62

The resulting $S$63 has ramification points, leaves, and bamboo points. Its tripod formula is

$S$64

A second function, the exponent,

$S$65

plays the same structural role as in the complex-analytic setting. If $S$66 is irreducible, its attaching point to $S$67 is

$S$68

Eggers decomposition relative to $S$69 is formulated for a product $S$70 with $S$71 irreducible. For each marked point $S$72, one sets $S$73 equal to the product of those factors $S$74 attached at $S$75. The decomposition is characterized by two equivalent conditions:

• $S$76: $S$77 factors only at marked points, and the intersection multiplicities $S$78 are prescribed by whether $S$79 is a leaf or not.
• $S$80: for every $S$81,

$S$82

where $S$83 is the product of irreducible factors of $S$84 attached at points above $S$85.

The central arithmetic condition is the Eggers condition: at a marked point $S$86,

$S$87

A weaker condition, the $S$88-condition, requires

$S$89

The paper proves that Eggers condition implies $S$90-condition, but for reducible $S$91 the converse may fail (Felipe et al., 28 Sep 2025).

The main criterion is exact: $S$92 for every marked point $S$93 of $S$94. If the condition holds at every marked point, then $S$95 holds for all $S$96; if the decomposition exists, then both the $S$97-condition and the Eggers condition hold at every marked point. In the squarefree case, the factor attached to any leaf is trivial (Felipe et al., 28 Sep 2025).

When the decomposition exists, the tree determines the data of the factors. For each marked point $S$98,

$S$99

where $L$00 is the set of direct successors of $L$01. For any irreducible factor $L$02,

$L$03

with the contact $L$04 computed from the function $L$05 on $L$06.

6. Examples, implications, and limitations

A worked complex-analytic example in the 2018 paper considers five branches with Newton–Puiseux series

$L$07

From these series one computes the characteristic exponents, the jumps of $L$08, and the coincidence orders

$L$09

$L$10

$L$11

The resulting tree has the ramification structure shown in Figure 1.1 of the paper, and the tripod formula yields intersection numbers such as

$L$12

This example exhibits the practical content of Eggers decomposition: once the tree and the functions $L$13 are known, mutual intersections are recovered directly (Barroso et al., 2018).

The positive-characteristic paper includes both success and failure cases. For

$L$14

with $L$15 and $L$16, the Eggers condition holds at the marked points, and

$L$17

admits Eggers decomposition, with the nontrivial factor attached at the unique ramification point. By contrast, if $L$18, then $L$19 equals the branch $L$20, the Eggers condition fails at the corresponding leaf, and decomposition breaks down. If $L$21, then $L$22 attaches at the root, the Eggers condition fails at the ramification point, and again decomposition fails (Felipe et al., 28 Sep 2025).

A sharper failure mechanism appears in characteristic $L$23 for

$L$24

Here the $L$25-condition holds at the ramification point because $L$26, but Eggers condition fails because $L$27. The polar is

$L$28

whose attaching point is an unmarked point $L$29 with $L$30. This is the paper’s explicit demonstration that $L$31-condition is necessary under decomposition but not sufficient (Felipe et al., 28 Sep 2025).

Several limitations are explicit. In positive characteristic, Puiseux factorization of $L$32 need not exist even when Eggers condition holds. The theory is developed for polars relative to $L$33 and trees built with respect to $L$34; analogous statements for $L$35 or for generic linear polars require separate analysis. In the complex-analytic setting, by contrast, the projective-limit theorem indicates that the valuative tree is the universal Eggers–Wall tree, so the dependence on a chosen finite curve germ is resolved by passage to $L$36 (Barroso et al., 2018).

Taken together, these results position Eggers decomposition at the intersection of equisingularity theory, valuation theory, and polar geometry. In characteristic zero and over $L$37, it provides a coordinate description of the valuative tree relative to an observer branch, with

$L$38

along the image of $L$39. In positive characteristic, it becomes a criterion-governed factorization theory for polar curves, where the nonvanishing modulo $L$40 of the quantities $L$41 is exactly what prevents inseparability pathologies from disrupting the canonical tree-indexed partition (Barroso et al., 2018, Felipe et al., 28 Sep 2025).