Projective Delineability: CAD and Beyond

• Projective delineability is defined in cylindrical algebraic decomposition as the decomposition of a polynomial’s projective zero set into graphs of continuous root functions with constant multiplicities.
• It enables robust handling of roots that pass through infinity by homogenizing polynomials, thereby bridging the gap between affine and projective analyses.
• Beyond CAD, the concept informs selection principles in topology and descriptive set theory by reformulating complex structures via image-based representations.

Searching arXiv for papers on “projective delineability” and closely related uses. Projective delineability most commonly denotes, in cylindrical algebraic decomposition (CAD), a projective-space analogue of classical delineability in which roots are tracked by continuous functions into the real projective line, allowing a root to pass through $\infty$ when an affine description would break down because the leading coefficient vanishes (Michel et al., 2024). The same phrase also appears in a broader mathematical register: in the study of selection principles it is used for “projective versions” of covering properties, where one requires that every second countable continuous image have a specified property (Osipov, 2018), while in descriptive set theory it is used informally for a graph-complexity phenomenon concerning total $\mathbf{\Pi}^1_3$-functions (Hoffelner, 20 May 2026). The dominant technical meaning, however, is the CAD notion.

1. Terminological scope

The phrase “projective delineability” is not uniform across the literature. In CAD it is a formal notion attached to polynomials and cell decompositions. In the Scheepers-diagram literature it denotes the systematic passage from a property of a space to the corresponding property of all second countable continuous images. In descriptive set theory it appears as an informal description of a definability phenomenon for graphs of total projective functions.

Area	Meaning	Representative formulation
CAD / single-cell construction	Projective-space analogue of delineability	Continuous projective root functions with constant multiplicity
Selection principles / Scheepers Diagram	Projective version of a topological property	Every second countable continuous image has property $\mathcal P$
Descriptive set theory	Informal graph-complexity “delineability”	Every total $\mathbf{\Pi}^1_3$-function has a $\mathbf{\Sigma}^1_3$ graph

This suggests that the shared terminology marks a recurring methodological move rather than a single invariant definition: an ambient property is reformulated after passage to a projective, image-based, or complexity-lowered representation. The CAD usage is the most explicit and theorematically developed instance.

2. CAD definition and projective-root geometry

In the CAD setting, the underlying geometric object is the real projective line. One presentation is

$\mathbb RP = \mathbb R^2/\!\sim,$

where $(x,y)\sim(\lambda x,\lambda y)$ for $\lambda\neq 0$, with affine embedding

$$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$

inverse

$\phi^{-1}(x:y)=\frac{x}{y},$

and distinguished point $\mathbf{\Pi}^1_3$0. An equivalent presentation used in the single-cell-construction paper is

$\mathbf{\Pi}^1_3$1

viewed as a circle (Michel et al., 2024, Nalbach et al., 1 Aug 2025).

Projective roots are defined through homogenization. For a univariate polynomial $\mathbf{\Pi}^1_3$2 of degree $\mathbf{\Pi}^1_3$3,

$\mathbf{\Pi}^1_3$4

A projective root is either a finite real root or the point at infinity, with multiplicity $\mathbf{\Pi}^1_3$5 in the latter case. For a polynomial $\mathbf{\Pi}^1_3$6 of degree $\mathbf{\Pi}^1_3$7 in the last variable and a base set $\mathbf{\Pi}^1_3$8, the projective zero set is

$\mathbf{\Pi}^1_3$9

The formal CAD definition requires decomposition of this projective zero set into graphs of continuous projective root functions. Specifically, $\mathcal P$0 is projectively delineable on $\mathcal P$1 if there exist $\mathcal P$2 and continuous functions $\mathcal P$3 such that

$\mathcal P$4

and for each $\mathcal P$5 there is a constant multiplicity $\mathcal P$6 realized at every base point. The single-cell-construction formulation is equivalent: for a cell $\mathcal P$7 and $\mathcal P$8, projective delineability on $\mathcal P$9 means that continuous functions $\mathbf{\Pi}^1_3$0 enumerate the projective roots of $\mathbf{\Pi}^1_3$1 for every $\mathbf{\Pi}^1_3$2, with distinct values and constant multiplicities. The analytic variant requires analytic root functions when the base is an analytic submanifold.

The key geometric intuition is that an affine root function may “escape” to infinity when the leading coefficient vanishes, but the corresponding projective root function remains continuous. In the affine picture, disappearance of a root at a degree-drop locus is singular behavior; in the projective picture, it is motion through $\mathbf{\Pi}^1_3$3.

3. Relation to classical delineability

Classical delineability is the affine version of the same graph-decomposition paradigm. For a cell $\mathbf{\Pi}^1_3$4 and polynomial $\mathbf{\Pi}^1_3$5, delineability means the real roots over $\mathbf{\Pi}^1_3$6 are exactly the graphs of continuous real-valued functions

$\mathbf{\Pi}^1_3$7

with

$\mathbf{\Pi}^1_3$8

and constant multiplicities. Classical CAD theory uses this to guarantee that sections and sectors behave regularly over the base cell (Michel et al., 2024).

The projective relation to the classical notion is subtle. The 2024 paper states that if the leading coefficient never vanishes on $\mathbf{\Pi}^1_3$9, then

$\mathbf{\Sigma}^1_3$0

and also that if the leading coefficient vanishes identically on a connected $\mathbf{\Sigma}^1_3$1 and $\mathbf{\Sigma}^1_3$2 is projectively delineable on $\mathbf{\Sigma}^1_3$3, then $\mathbf{\Sigma}^1_3$4 is delineable on $\mathbf{\Sigma}^1_3$5. It further states that neither converse implication holds in general: delineability does not necessarily imply projective delineability, and projective delineability does not necessarily imply delineability. The single-cell-construction paper isolates the practically most important sufficient bridge:

If $\mathbf{\Sigma}^1_3$6 is projectively delineable on $\mathbf{\Sigma}^1_3$7, and the leading coefficient is sign-invariant on $\mathbf{\Sigma}^1_3$8, then $\mathbf{\Sigma}^1_3$9 is delineable on $\mathbb RP = \mathbb R^2/\!\sim,$0. This is precisely the regime in which projective delineability can be used as a computational surrogate for classical delineability without sacrificing the affine conclusions needed later (Michel et al., 2024, Nalbach et al., 1 Aug 2025).

The main local theorem of the 2024 paper shows that if $\mathbb RP = \mathbb R^2/\!\sim,$1 is a connected analytic submanifold of $\mathbb RP = \mathbb R^2/\!\sim,$2, $\mathbb RP = \mathbb R^2/\!\sim,$3 never nullifies on $\mathbb RP = \mathbb R^2/\!\sim,$4, $\mathbb RP = \mathbb R^2/\!\sim,$5 is not the zero polynomial, and $\mathbb RP = \mathbb R^2/\!\sim,$6 is order-invariant on $\mathbb RP = \mathbb R^2/\!\sim,$7, then for each $\mathbb RP = \mathbb R^2/\!\sim,$8 there exists a connected neighbourhood $\mathbb RP = \mathbb R^2/\!\sim,$9 such that $(x,y)\sim(\lambda x,\lambda y)$0 is projectively delineable on $(x,y)\sim(\lambda x,\lambda y)$1, and $(x,y)\sim(\lambda x,\lambda y)$2 is order-invariant on each projective $(x,y)\sim(\lambda x,\lambda y)$3-section over $(x,y)\sim(\lambda x,\lambda y)$4. Under the additional hypothesis that $(x,y)\sim(\lambda x,\lambda y)$5 is simply connected, the global theorem upgrades this to projective delineability on all of $(x,y)\sim(\lambda x,\lambda y)$6. The necessity of the extra topological hypothesis is witnessed by an explicit connected analytic submanifold $(x,y)\sim(\lambda x,\lambda y)$7 and a degree-$(x,y)\sim(\lambda x,\lambda y)$8 polynomial $(x,y)\sim(\lambda x,\lambda y)$9 satisfying the local discriminant hypotheses but failing to be projectively delineable globally. A common misconception is therefore that projective delineability is merely “ordinary delineability with the leading coefficient removed”; the published results instead treat it as a distinct condition whose interaction with classical delineability depends on leading-coefficient behavior and on the topology of the base.

4. Single-cell construction, projection strategy, and empirical profile

The algorithmic motivation is single cell construction in CAD-based quantifier elimination and SMT solving. The 2025 paper places projective delineability in the context of exploration-guided CAD technology, including NLSAT, NuCAD, and CAlC, all of which use single cell construction. The classical loop adds a non-nullifying coefficient, leading coefficients and discriminants for delineability, and resultants to preserve order relations among relevant root functions. The projective adaptation changes only part of this logic: delineability is kept for the polynomials defining the cell bounds, while projective delineability is used for the remaining polynomials (Nalbach et al., 1 Aug 2025).

Operationally, the modified algorithm adds a coefficient to avoid nullification, adds only the discriminant / order-invariance polynomial for projective delineability, keeps the discriminant / order-invariance polynomials for the bound-defining polynomials to ensure ordinary delineability there, optionally omits leading coefficients for rootless polynomials via an optimization lemma, and replaces linear ordering arguments by a cyclic ordering relation on projective roots. The crucial sign-invariance theorem states that, if $\lambda\neq 0$0 is projectively delineable on a connected base $\lambda\neq 0$1, if $\lambda\neq 0$2 and $\lambda\neq 0$3 are delineable bound-defining polynomials with root functions $\lambda\neq 0$4, and if each projective root function of $\lambda\neq 0$5 is either identical to a bound or cyclically ordered away from the interval at the sample point, then $\lambda\neq 0$6 is sign-invariant on the corresponding cylinder or section. The transitivity substitute is likewise cyclic: if

$\lambda\neq 0$7

and the four projective root functions remain pairwise distinct on $\lambda\neq 0$8, then $\lambda\neq 0$9 on $$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$0.

The computational point is that some leading coefficients can be omitted because projective delineability allows roots to pass through $$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$1. The paper is explicit that this benefit is conditional. If the cell is unbounded on one side, or if a relevant root may cross $$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$2, then the leading coefficient may still be necessary; in other situations a resultant can replace it; and when the polynomial is root-free on the cell, no additional polynomial may be needed. This produces a flexible projection strategy rather than a monotone reduction rule.

The implementation in SMT-RAT was evaluated on the SMT-LIB QF_NRA benchmark set of $$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$3 instances with a $$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$4 timeout and $$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$5 memory limit. Four variants were tested: BC, LDB, BC-PD, and LDB-PD. The paper reports that projective delineability does not significantly change the overall number of solved instances and does not significantly improve aggregate runtime or intermediate cell quality. About $$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$6–$$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$7 instances are solved by both baseline and PD variants, only a small number are solved exclusively by one variant or the other, and roughly $$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$8–$$\phi:\mathbb R\to \mathbb RP,\qquad x\mapsto (x:1),$$9 are solved by neither. Among instances solved by BC-PD, the leading coefficient can be omitted for $\phi^{-1}(x:y)=\frac{x}{y},$0 polynomials, the optimization cannot be applied for $\phi^{-1}(x:y)=\frac{x}{y},$1 polynomials because the cell is unbounded in some direction, and for $\phi^{-1}(x:y)=\frac{x}{y},$2 polynomials it cannot be applied because there is no root on both sides of the bounds or no suitable replacing resultant was found. On solved BC-PD instances, about $\phi^{-1}(x:y)=\frac{x}{y},$3 of algebraic computation time is spent on discriminants, about $\phi^{-1}(x:y)=\frac{x}{y},$4 on resultants, and almost nothing on coefficients; among the maximum-total-degree projection polynomials in an instance, only $\phi^{-1}(x:y)=\frac{x}{y},$5 are coefficients, versus $\phi^{-1}(x:y)=\frac{x}{y},$6 discriminants and $\phi^{-1}(x:y)=\frac{x}{y},$7 resultants. The experimental conclusion is therefore restrained: projective delineability is mathematically and algorithmically useful, but leading coefficients are usually not the dominant bottleneck.

5. Projective versions in the Scheepers diagram

In topology, the projective vocabulary is organized around Arhangel’skii’s definition:

A space $\phi^{-1}(x:y)=\frac{x}{y},$8 is projectively $\phi^{-1}(x:y)=\frac{x}{y},$9 if every second countable continuous image of $\mathbf{\Pi}^1_3$00 has property $\mathbf{\Pi}^1_3$01. Because every second countable space embeds into $\mathbf{\Pi}^1_3$02, this is equivalent to requiring that every continuous image $\mathbf{\Pi}^1_3$03 have the relevant property. The 2018 paper develops functional characterizations of the projective versions of the selection principles in the Scheepers Diagram and identifies these “projective versions” with projective delineability in that context (Osipov, 2018).

The properties treated include Rothberger $\mathbf{\Pi}^1_3$04, Menger $\mathbf{\Pi}^1_3$05, Hurewicz $\mathbf{\Pi}^1_3$06, Gerlits–Nagy $\mathbf{\Pi}^1_3$07, as well as $\mathbf{\Pi}^1_3$08, $\mathbf{\Pi}^1_3$09, $\mathbf{\Pi}^1_3$10, $\mathbf{\Pi}^1_3$11, $\mathbf{\Pi}^1_3$12, $\mathbf{\Pi}^1_3$13, and $\mathbf{\Pi}^1_3$14. The guiding pattern is uniform: each projective property is characterized by images into $\mathbf{\Pi}^1_3$15, by a cozero-cover selection principle on $\mathbf{\Pi}^1_3$16, or by a function-space property of $\mathbf{\Pi}^1_3$17.

Projective property	Criterion on $\mathbf{\Pi}^1_3$18	Criterion on $\mathbf{\Pi}^1_3$19
Rothberger	$\mathbf{\Pi}^1_3$20	$\mathbf{\Pi}^1_3$21
Menger	$\mathbf{\Pi}^1_3$22	$\mathbf{\Pi}^1_3$23
Hurewicz	$\mathbf{\Pi}^1_3$24	$\mathbf{\Pi}^1_3$25
Gerlits–Nagy	$\mathbf{\Pi}^1_3$26	$\mathbf{\Pi}^1_3$27

The paper proves, for example, that

$\mathbf{\Pi}^1_3$28

$\mathbf{\Pi}^1_3$29

and

$\mathbf{\Pi}^1_3$30

For projective Hurewiczness, the paper singles out

$\mathbf{\Pi}^1_3$31

together with the corresponding $\mathbf{\Pi}^1_3$32 characterization. The article also records preservation and small-cardinal facts, such as: spaces of cardinality $\mathbf{\Pi}^1_3$33 are projectively Rothberger; spaces of cardinality $\mathbf{\Pi}^1_3$34 are projectively Menger; spaces of cardinality $\mathbf{\Pi}^1_3$35 are projectively Hurewicz; spaces of cardinality $\mathbf{\Pi}^1_3$36 are projectively $\mathbf{\Pi}^1_3$37. In this literature, “projective delineability” therefore does not refer to roots in projective space; it refers to the possibility of reading off image-based covering properties from cozero-cover principles and from the selection structure of $\mathbf{\Pi}^1_3$38.

6. Descriptive-set-theoretic analogy and cross-field significance

A third usage appears in the 2026 paper on graphs of total projective functions. Its central theorem states that, assuming $\mathbf{\Pi}^1_3$39, there is a model of $\mathbf{\Pi}^1_3$40 in which every total $\mathbf{\Pi}^1_3$41-function has a $\mathbf{\Pi}^1_3$42 graph, and the strengthened model also satisfies $\mathbf{\Pi}^1_3$43-reduction while $\mathbf{\Pi}^1_3$44-uniformization fails. The paper explicitly says that it is best viewed as a “delineability result for projective functions,” not as the CAD notion of projective delineability (Hoffelner, 20 May 2026).

The mechanism is definability-theoretic rather than geometric. A total $\mathbf{\Pi}^1_3$45-function is one whose graph is a boldface $\mathbf{\Pi}^1_3$46 subset of $\mathbf{\Pi}^1_3$47, and the paper constructs a forcing extension in which each such graph is also $\mathbf{\Pi}^1_3$48. The resulting principle is incompatible with $\mathbf{\Pi}^1_3$49-uniformization and hence with the usual projective-determinacy picture. The key obstruction is a total $\mathbf{\Pi}^1_3$50 relation $\mathbf{\Pi}^1_3$51 with full domain that cannot be uniformized by any function whose graph is $\mathbf{\Pi}^1_3$52.

This descriptive-set-theoretic usage is terminologically adjacent but conceptually distinct. It does not introduce projective root functions, projective zero sets, or cozero-cover selection principles. Instead, it treats “delineability” as a graph-complexity simplification: a total projective function becomes representable by a graph of lower projective complexity. This suggests a broad family resemblance among the three literatures: in each case, structure that is difficult to control directly becomes tractable after passage to an auxiliary representation—projective roots in $\mathbf{\Pi}^1_3$53, second countable images in $\mathbf{\Pi}^1_3$54, or lower-complexity graph codes.

Across these domains, the CAD meaning remains the canonical technical sense of projective delineability. There it names a precise replacement for affine delineability when roots may pass through infinity, with local and global existence theorems, a projective-to-classical bridge under leading-coefficient hypotheses, and concrete algorithmic consequences for single-cell CAD construction. The topological and descriptive-set-theoretic usages broaden the phrase into an organizing metaphor for image-based or projective-level reformulations, but they do not erase the specificity of the CAD definition.