Pfaffian function

In mathematics, the pfaffian functions are a certain class of functions introduced by Khovanskii in the 1970s. They are named after German mathematician Johann Pfaff.

Basic Definition

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, f(x) = e^x. If we differentiate this function we get e^x again, that is

$f^{\prime }(x)=f(x)$.

Another example of a function like this is the reciprocal function, g(x) = 1/x. If we differentiate this function we will see that

$g^{\prime }(x)=-g(x)^{2}.$

Other functions may not have the above property, but their derivative may be written in terms of functions like those above. For example if we take the function h(x) = e^xlog(x) then we see

$h^{\prime }(x)=e^{x}\log x+{\frac {1}{x}}e^{x}=h(x)+f(x)g(x).$

Functions like these form the links in a so-called pfaffian chain. Such a chain is a sequence of functions, say f₁, f₂, f₃, etc, 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 proceeding it in the chain (specifically as a polynomial in those functions and the variables involved). So with the functions above we have that f, g, h is a pfaffian chain.

A pfaffian function is then just a function comprised only of those functions appearing in a pfaffian chain, plus standard polynomials in the variable. So with the pfaffian chain just mentioned, functions like F(x) = x³f(x)²‒ 2g(x)h(x) are pfaffian.

Rigorous Definition

Let U be an open domain in Rⁿ. A pfaffian chain of order r ≥ 0 and degree α ≥ 1 in U is a sequence of real analytic functions f₁,…, f_r in U satisfying differential equations

${\frac {\partial f_{i}}{\partial x_{j}}}=P_{i,j}({\boldsymbol {x}},f_{1}({\boldsymbol {x}}),\ldots ,f_{i}({\boldsymbol {x}}))$

for i = 1,…,r where P_i,j ∈ R[x₁,...,x_n,y₁,...,y_i] are polynomials of degree ≤ α. A function f on U is called a pfaffian function of order r and degree (α,β) if

$f({\boldsymbol {x}})=P({\boldsymbol {x}},f_{1}({\boldsymbol {x}}),\ldots ,f_{r}({\boldsymbol {x}})),\,$

where P ∈ R[x₁,...,x_n,y₁,...,y_r] is a polynomial of degree at most β ≥ 1.

Examples

1. Any polynomial is a pfaffian function with r = 0.
2. The function f(x)=e^x is pfaffian with r = 1 and α = β = 1 due to the equation f ′ = f.
3. The algebraic functions are pfaffian.
4. Any combination of polynomials, exponentials, the trigonometric functions on bounded intervals, and their inverses, in any finite number of variables, is pfaffian.

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:

${\begin{aligned}f_{1}^{\prime }&=P_{1}(x,f_{1})\\f_{2}^{\prime }&=P_{2}(x,f_{1},f_{2})\\f_{3}^{\prime }&=P_{3}(x,f_{1},f_{2},f_{3}),\end{aligned}}$

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. So, for example, a Noetherian chain of order three is composed of three functions f₁, f₂, f₃, satisfying the equations

${\begin{aligned}f_{1}^{\prime }&=P_{1}(x,f_{1},f_{2},f_{3})\\f_{2}^{\prime }&=P_{2}(x,f_{1},f_{2},f_{3})\\f_{3}^{\prime }&=P_{3}(x,f_{1},f_{2},f_{3}).\end{aligned}}$

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

Any pfaffian chain is also a Noetherian chain, the extra variables in each polynomial are simply redundant in this case. But not every Noetherian chain is pfaffian. If we take f₁(x) = sin(x) and f₂(x) = cos(x) then we have the equations

${\begin{aligned}f_{1}^{\prime }(x)&=f_{2}(x)\\f_{2}^{\prime }(x)&=-f_{1}(x),\end{aligned}}$

and these hold for all real numbers x, so f₁,f₂ is a noetherian chain on all of R. But there is no polynomial P(x,y) such that the derivative of sin(x) can be written as P(x,sin(x)), and so this chain is not pfaffian.

References

• A.G. Khovanskii, Fewnomials, Princeton University Press, 1991.