Existential theory of the reals¶

The set and associated decision problem of true existential sentences formed from polynomial equalities and inequalities over real variables.

Version: v2 · 2026-09-19 · History
Domain-specific #: 4468
Domain group: Applied Sciences & Engineering
Origin domain: Computer Science & Software Engineering
Subdomain: Computational Complexity → Computer Science & Software Engineering
Aliases: ETR

Core Idea¶

Only existential quantifiers and quantifier-free real-polynomial constraints define the canonical fragment, encoding model affects bit complexity and the complexity class exists between NP and PSPACE without being known equal to either.^[1] A finite system of real algebraic constraints is encoded as one existential formula; decision asks whether the corresponding semialgebraic set is nonempty, using real-algebraic elimination or geometric reasoning. The abstraction is therefore identified by a declared carrier, a transformation or constraint over that carrier, and an invariant that tells an analyst whether the named structure is genuinely present.

The load-bearing residual is not the broad topic of computational complexity. It is the domain-specific identity fixed by the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit. That residual remains recognizable when examples, notation, scale, or implementation change, but it disappears if the carrier is mistyped, the condition that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test. This gives the entry an operational identity rather than merely a historical label.

A useful analysis keeps three layers separate. The constitutive layer says what must be true: the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit. The evidential layer asks what observation or proof warrants the claim: type the carrier, state every parameter and convention in the definition, test that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases. The use layer asks what reasoning becomes available once the identity is established: recognizing and comparing instances of Existential theory of the reals, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Conflating the layers is the most common source of scope inflation.

How would you explain it like I'm…

Can Any Numbers Work?

Imagine a puzzle with rules like "this number times itself is more than that number." The question is: can you find any numbers at all, even ones with endless decimals, that follow every rule at once? Computer scientists study how hard that yes-or-no question is to answer.

The Real-Number Rule Puzzle

Some puzzles give you a list of rules about unknown numbers. The rules can include multiplying numbers together, and they can say things like "equals," "is less than," "and," and "or." The Existential theory of the reals asks one question: does at least one choice of numbers, including decimals that never end, make all the rules true? You don't have to find the best answer, only decide whether any answer exists. Computer scientists study how hard it is for a computer to decide this.

Solvability of Real Polynomial Systems

The Existential theory of the reals consists of true statements of the form "there exist real numbers x1,...,xn such that" some combination (using AND, OR, NOT) of polynomial equations and inequalities holds. Deciding such a statement asks whether the set of solutions—a semialgebraic set—is nonempty. Only "there exists" is allowed; statements mixing in "for all" belong to the larger full theory of the real numbers. The decision problem is at least as hard as NP and no harder than PSPACE, but it isn't known to equal either. Problems that are equally hard are called ETR-complete, and how the coefficients are written in bits matters for measuring the difficulty.

The Existential theory of the reals is the fragment of the first-order theory of the real numbers consisting of sentences ∃x₁…∃xₙ φ. Here φ is a quantifier-free Boolean combination of polynomial equations and strict or weak inequalities with integer or rational coefficients. Truth of such a sentence is equivalent to nonemptiness of a semialgebraic set. The decision problem takes a finite system of real algebraic constraints, encodes it as one existential formula, and asks whether it is satisfiable. Coefficient encoding matters because complexity is measured in bits. The associated complexity class ∃ℝ contains NP and is contained in PSPACE, and neither containment is known to be equality. Problems are ETR-hard or ETR-complete under specified polynomial-time reductions. Decision procedures use real-algebraic elimination or geometric arguments. ETR must be kept distinct from the full first-order theory of real closed fields, which allows quantifier alternation and has different complexity.

Structural Signature¶

Carrier: the typed computational complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets
Inputs or antecedent state: the exact computational complexity carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Existential theory of the reals
Constitutive operation: A finite system of real algebraic constraints is encoded as one existential formula; decision asks whether the corresponding semialgebraic set is nonempty, using real-algebraic elimination or geometric reasoning.
Invariant: the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit
Recognition test: type the carrier, state every parameter and convention in the definition, test that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases
Output or consequence: recognizing and comparing instances of Existential theory of the reals, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions
Failure boundary: the carrier is mistyped, the condition that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test

What It Is Not¶

It is not the whole field of computational complexity. The field contains many questions and methods that do not instantiate Existential theory of the reals.
It is not its most familiar example. A canonical instance directly demonstrates that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit. exhibits the structure, but the example is evidence for the abstraction rather than its definition.
It is not the neighboring catalog concept Theory of real closed fields. The full theory permits alternating quantifiers and admits quantifier elimination; ETR is its existential fragment and the associated feasibility complexity class.
It is not a claim that every boundary case has one uncontested classification. a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Existential theory of the reals must control the decision
It is not an unrestricted metaphor for any process that seems similar. Outside computational complexity, the vocabulary and validity conditions do not transfer literally.

Scope of Application¶

Existential theory of the reals belongs to computational complexity and is useful where the analyst can specify the typed computational complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, then evaluate the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit. The scope is broad within that domain but bounded by the need for the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit. The entry records a descriptive analytical identity; practical use requires the governing domain's evidence, standards, and safety obligations.^[2]

Definition and recognition. Determine whether a proposed instance satisfies the constitutive conditions rather than merely sharing terminology.
Construction or evolution. Track how the exact computational complexity carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Existential theory of the reals are converted, constrained, or organized by A finite system of real algebraic constraints is encoded as one existential formula; decision asks whether the corresponding semialgebraic set is nonempty, using real-algebraic elimination or geometric reasoning..
Comparison. Compare instances using carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior, without treating convenience measures as the definition.
Boundary analysis. Diagnose cases where a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Existential theory of the reals must control the decision and state which convention or theorem controls the decision.
Downstream reasoning. Use the established identity to support recognizing and comparing instances of Existential theory of the reals, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions while preserving the assumptions under which the inference is valid.

Clarity¶

The abstraction clarifies a crowded vocabulary by making the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit the center of the account. A claim should name the carrier, the governing operation or relation, the applicable assumptions, and the recognition test. A bare label is insufficient because the name Existential theory of the reals can be used for a formal identity, an implementation, or a neighboring result unless carrier and convention are stated. The disciplined statement is: given the exact computational complexity carrier, defining parameters and conventions, boundary conditions, source evidence, comparison cases, and any measurement or proof assumptions needed to evaluate Existential theory of the reals, the structure counts as Existential theory of the reals exactly when the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit.

This format also separates identity from measurement. Empirical, computational, or documentary proxies support recognition only under declared validity and uncertainty assumptions; formal cases require proof rather than measurement. Measurements can be noisy, implementations can approximate, and proofs can use equivalent characterizations; none of those facts licenses changing the object being measured. When reports disagree, first check scope and convention, then data or proof, and only then interpret the disagreement as substantive.

Manages Complexity¶

Without the abstraction, an analyst must reason directly over many local details: the carrier roles, admissibility assumptions, competing conventions, derived invariants, boundary cases, and proof or validation obligations specific to Existential theory of the reals. Existential theory of the reals compresses them into the roles in the structural signature. That compression permits comparison across instances without erasing the variables that determine validity. It also exposes which details may be varied safely and which are constitutive.

The compression has a price. A single label can hide canonical, generalized, restricted, approximate, computational, empirical, and historically variant formulations of Existential theory of the reals. Good use therefore carries a small declaration of assumptions alongside the name. The abstraction manages complexity when it reduces the state space of the question while keeping the failure boundary visible; it mismanages complexity when the label substitutes for that boundary analysis.

Abstract Reasoning¶

Identify the carrier. State what the elements, states, objects, or observations are: the typed computational complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets. Reject examples whose alleged carrier belongs to a different problem.
Lock the constitutive rule. Express the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit independently of one notation or implementation. This step prevents the canonical example from becoming the definition.
Derive consequences. From the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit, infer recognizing and comparing instances of Existential theory of the reals, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions. Record each assumption used so that a later change of setting does not silently preserve an invalid conclusion.
Test adversarial cases. Examine a generalized or degenerate case may change existence, uniqueness, measurement, or naming conventions, so the exact definition of Existential theory of the reals must control the decision and an object that resembles Existential theory of the reals in purpose or vocabulary but does not satisfy its invariant is outside the class. A robust identity explains why the first is convention-sensitive and why the second is outside the class.
Compare and refine. Use carrier, parameters, convention, domain, scale, boundary conditions, evidence, exact versus approximate form, and limiting behavior to compare legitimate instances, and refine the model when discrepancies reflect hidden variation rather than failure of the abstraction itself.

Knowledge Transfer¶

Knowledge transfers strongly among subfields of computational complexity because they reuse the typed computational complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets, A finite system of real algebraic constraints is encoded as one existential formula; decision asks whether the corresponding semialgebraic set is nonempty, using real-algebraic elimination or geometric reasoning., and type the carrier, state every parameter and convention in the definition, test that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases. A theorem, diagnostic, or modeling warning can travel when those roles remain literal. For example, the distinction between constitutive identity and a convenient observable transfers from A canonical instance directly demonstrates that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit. to An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation..^[3]

Transfer outside the home domain is weaker. The skeletal pattern—type the carrier, apply the defining mechanism of Existential theory of the reals, preserve its invariant, and derive only consequences licensed by the stated boundary—may suggest an analogy, but the domain-specific mechanisms, admissible evidence, and consequences do not come along automatically. The safe transfer procedure maps each role explicitly, checks the invariant again, and refuses the name when only a superficial resemblance remains.

Examples¶

Canonical¶

A canonical instance directly demonstrates that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit. The example exposes the carrier and directly tests that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit; changing incidental notation preserves the identity, while removing that condition destroys it. This example is canonical because every role can be inspected: the carrier is the typed computational complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets; the operative rule is A finite system of real algebraic constraints is encoded as one existential formula; decision asks whether the corresponding semialgebraic set is nonempty, using real-algebraic elimination or geometric reasoning.; the invariant is the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit; and the result supports recognizing and comparing instances of Existential theory of the reals, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions.^[1] Changing incidental notation or scale leaves the structure intact, while removing the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit destroys the classification.

Mapped back: the typed computational complexity carrier, including objects, relations, parameters, conventions, evidence, boundaries, and comparison targets → A finite system of real algebraic constraints is encoded as one existential formula; decision asks whether the corresponding semialgebraic set is nonempty, using real-algebraic elimination or geometric reasoning. → the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit → recognizing and comparing instances of Existential theory of the reals, deriving its domain-specific consequences, selecting valid models or methods, and preventing transfer beyond its assumptions

Applied / In Practice¶

An applied instance preserves the invariant under changed notation, scale, dataset, jurisdiction, or implementation. The applied case qualifies only because the same invariant and boundary test remain literal under changed parameters or implementation. The applied case is not licensed merely by vocabulary. It qualifies because the same recognition test—type the carrier, state every parameter and convention in the definition, test that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit, compare the nearest accepted identity, and report counterexamples, uncertainty, and limiting cases—can be run and because the same failure boundary—the carrier is mistyped, the condition that the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields are explicit fails, a neighboring object is substituted, or notation and topical resemblance replace the constitutive test—remains meaningful.^[2] The case also shows why practical outputs should report assumptions, resolution, and uncertainty instead of a naked label.

Mapped back: declared instance → recognition test → boundary check → qualified use

Structural Tensions¶

T1: Axiomatic identity vs. operational recognition. The defining conditions may be exact while empirical or computational recognition is approximate. Neither pole can be removed without changing the analytical task. Diagnostic: Can the reviewer state both the exact condition and the evidence used to infer it?
T2: Local roles vs. global consequence. The mechanism is enacted through local relations, but the abstraction is usually valued for a global classification or prediction. Neither pole can be removed without changing the analytical task. Diagnostic: Does the claimed global result actually follow from the declared local conditions?
T3: Ideal form vs. finite representation. Theory states a clean invariant while data structures, measurements, or proofs expose only finite representations. Neither pole can be removed without changing the analytical task. Diagnostic: Would increasing resolution converge toward the same classification?
T4: Canonical convention vs. legitimate variants. A standard formulation supports communication, while variants may preserve the same core under changed assumptions. Neither pole can be removed without changing the analytical task. Diagnostic: Which role is invariant across variants, and which convention-specific conclusion changes?
T5: Compression vs. hidden assumptions. The name compresses a complex argument but can conceal prerequisites. Neither pole can be removed without changing the analytical task. Diagnostic: Can each downstream inference be traced to an explicit assumption?
T6: Autonomous residual vs. reduction to catalog neighbors. The candidate uses broader structures but adds an identity-bearing residual. Neither pole can be removed without changing the analytical task. Diagnostic: After subtracting the proposed parent and named neighbors, does the constitutive residual still support independent diagnostics?

Structural–Framed Character¶

The entry is structurally mixed but domain-framed. Its portable skeleton is type the carrier, apply the defining mechanism of Existential theory of the reals, preserve its invariant, and derive only consequences licensed by the stated boundary. Its identity-bearing terms—Existential theory of the reals, carrier, parameter, invariant, boundary, evidence, model, transformation, and application—derive their meaning from computational complexity and cannot be replaced by generic systems language without losing the tests that distinguish valid from invalid instances.

This mixed character explains why the abstraction is reusable inside the domain yet does not meet the Prime bar. The structure organizes reasoning, but its claims still depend on domain-specific objects, evidence, and intervention semantics.

Structural Core vs. Domain Accent¶

The structural core consists of a carrier, A finite system of real algebraic constraints is encoded as one existential formula; decision asks whether the corresponding semialgebraic set is nonempty, using real-algebraic elimination or geometric reasoning., a recognition invariant, and a consequence. That skeleton may resemble patterns elsewhere, especially type the carrier, apply the defining mechanism of Existential theory of the reals, preserve its invariant, and derive only consequences licensed by the stated boundary. The domain accent is not decorative: Existential theory of the reals, carrier, parameter, invariant, boundary, evidence, model, transformation, and application determine what counts as an admissible carrier, a valid transition, and successful evidence.

The abstraction therefore remains domain-specific. A cross-domain reuse that preserves only words such as 'balance,' 'cut,' 'sequence,' 'loss,' or 'simulation' is metaphor. Literal transfer requires the original role structure and diagnostics, which in this case remain anchored in computational complexity.

This entry is a kind of Verification.

It remains its own entry because its identity is fixed by the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields.

Relationships to Other Abstractions¶

Current abstraction Existential theory of the reals Domain-specific

Parents (1) — outgoing typed relations

Existential theory of the reals is a kind of Verification Prime

It remains its own entry because its identity is fixed by the real variables and coefficient encoding, existential quantifier block, quantifier-free Boolean combination, polynomial equations strict or weak inequalities, real-number interpretation, satisfiability or semialgebraic nonemptiness, reduction model, decision algorithm and complexity bounds, ETR-hardness and completeness and distinction from full first-order theory of real closed fields.

Hierarchy path (1) — routes to 1 parentless root

Existential theory of the reals → is a kind of Verification → is a kind of Evaluation → presupposes Comparison

Neighborhood in Abstraction Space¶

Existential theory of the reals sits in a crowded region of the domain-specific corpus (18^th percentile for distinctiveness): several abstractions share nearly its structure, so a description that fits it tends to fit its neighbors too.

Family — Computational Complexity Classes & Hierarchies (30 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-25

Not to Be Confused With¶

Theory of real closed fields. The full theory permits alternating quantifiers and admits quantifier elimination; ETR is its existential fragment and the associated feasibility complexity class.
One canonical example. An instance demonstrates the structure but does not define the whole abstraction.
Measurement or implementation of Existential theory of the reals. A proxy or realization is evidence for the abstraction, not the abstraction itself.
Generalized Existential theory of the reals. An extension qualifies only when its changed axioms and retained invariant are stated.

References¶

[1] Saugata Basu, Richard Pollack, Marie-Françoise Roy, 'Algorithms in Real Algebraic Geometry', Springer-Verlag, 2006, doi:10.1007/3-540-33099-2_14. registry ↩a ↩b Show verification details

[2] John Canny, '[[Symposium on Theory of Computing', ACM, 1988, doi:10.1145/62212.62257. registry ↩a ↩b

[3] Yu. V Matiyasevich, 'Mathematical events of the twentieth century', Springer-Verlag, 2006, doi:10.1007/3-540-29462-7_10. registry ↩