Wilkie's theorem¶
In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
Core Idea¶
Wilkie's theorem is treated here as the recurring mathematicslogicstatistics identity summarized by this source-grounded definition: In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties. In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties. Wilkie's theorem states that there is an integer n\geq m and polynomials.
Scope of Application¶
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Intermediate results. Gabrielov's theorem applies to the real field with all restricted analytic functions adjoined, whereas Wilkie's theorem removes the need to restrict the function, but only allows one to add the exponential.
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Formulations. In terms of model theory, Wilkie's theorem deals with L{\textrm{exp}} = (+,-,\cdot, , the language of ordered rings with an exponential function e^x .
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Formulations. This result proves that the theory of the structure \R{\textrm{exp}} , the real ordered field with the exponential function, is model complete.
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Gabrielov's theorem. This earlier theorem of Andrei Gabrielov dealt with sub-analytic sets, or the language L{\textrm{an}} of ordered rings with a function symbol for each proper analytic function on \R^m.
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Gabrielov's theorem. Hence the theory of the real ordered field with restricted analytic functions is model complete.
Clarity¶
A clear use of Wilkie's theorem names the carrier, the operative relation, and the conditions under which the source treats the identity as present. The minimal definition is In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
Manages Complexity¶
Wilkie's theorem compresses multiple mathematicslogicstatistics details into a stable diagnostic relation. The source shows both the central mechanism—in mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.—and the practical consequence—thus, while this theory does not have full quantifier elimination, formulae can be put in a particularly simple form.
Abstract Reasoning¶
- Type the carrier. Identify the mathematicslogicstatistics entities to which the claim applies.
- State the relation. Use the source-grounded identity: In mathematics, Wilkie's theorem is a result by Alex Wilkie about the theory of ordered fields with an exponential function, or equivalently about the geometric nature of exponential varieties.
- Check operation and conditions. In terms of model theory, Wilkie's theorem deals with L{\textrm{exp}} = (+,-,\cdot, , the language of ordered rings with an exponential function e^x .
- Demand recognition evidence.
Knowledge Transfer¶
Within the home domain. Knowledge about Wilkie's theorem transfers literally when a new case preserves the same carrier type, relation, and recognition test. Gabrielov's theorem applies to the real field with all restricted analytic functions adjoined, whereas Wilkie's theorem removes the need to restrict the function, but only allows one to add the exponential function. In terms of model theory, Wilkie's theorem deals with L{\textrm{exp}} = (+,-,\cdot, , the language of ordered rings with an exponential function e^x . Beyond the home domain. No canonical parent is asserted for Wilkie's theorem.
Neighborhood in Abstraction Space¶
Wilkie's theorem sits in a sparse region of the domain-specific corpus (85th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.
Family — Named Analytic Theorems & Operators (39 abstractions)
Nearest neighbors
- Egorov's theorem — 0.83
- Minakshisundaram–Pleijel zeta function — 0.81
- Functional determinant — 0.81
- Exponential polynomial — 0.81
- Scattered order — 0.81
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