In mathematics, Pfaffian functions are a certain class of functionss whose derivative can be written in terms of the original function. They were originally introduced by Askold Khovanskii in the 1970s, but are named after German mathematician Johann Pfaff.

Motivation

Some functions, when differentiated, give a result which can be written in terms of the original function. Perhaps the simplest example is the exponential function again; that is,

Another example is the reciprocal. Differentiating this function, we see that

, we have that

Functions like these form the links in a so-called Pfaffian chain. Such a chain is a sequence with the property that if we differentiate any of the functions in this chain then the result can be written in terms of the function itself and all the functions preceding it in the chain (specifically as a polynomial is a Pfaffian chain.

A Pfaffian function are Pfaffian.

Formal definition

be an open. A Pfaffian chain is a sequence of real analytic functions satisfying differential equations

are polynomials of degree is called a Pfaffian function if

are collectively known as the format of the Pfaffian function, and give a useful measure of its complexity.

satisfying the differential equations

where

.

Examples

• .

• .

• .

• All of algebraic functions are Pfaffian on suitable domains, as are hyperbolic functions. Trigonometric functions on bounded intervals.

• In fact, all elementary functions and Liouvillian functions are Pfaffian.

In model theory

, the ordered field of real numbers. Andrei Gabrielov is model complete. was just the projection of some higher-dimensional set defined by identities and inequalities involving these restricted analytic functions.

Alex Wilkie showed that one has the same result if instead of adding every restricted analytic function, one just adds the unrestricted, a result known as Wilkie's theorem. would give the same result. In particular one may add all as a variant of Gabrielov's result. The result on exponentiation is not a special case of this result (even though exp is a Pfaffian chain by itself), as it applies to the unrestricted exponential function.

Closely related to model-completeness are the results on o-minimality is an o-minimal structure, whereas van den Dries and Miller combining restricted analytic functions and the unrestricted exponential function is o-minimal.

Noetherian functions

The equations above that define a Pfaffian chain are said to satisfy a triangular condition, since the derivative of each successive function in the chain is a polynomial in one extra variable. Thus if they are written out in turn a triangular shape appears:

and so on. If this triangularity condition is relaxed so that the derivative of each function in the chain is a polynomial in all the other functions in the chain, then the chain of functions is known as a Noetherian chain, and a function constructed as a polynomial in this chain is called a Noetherian function., satisfying the equations

The name stems from the fact that the ring generated by the functions in such a chain is Noetherian.

then we have the equations

, and so this chain is not Pfaffian.

Notes

References