Pfaffian function

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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.

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[edit] 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)=ex. If we differentiate this function we get ex 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, when differentiated, they may be written in terms of functions like those above. For example if we take the logarithm function h(x)=log(x) then we see

h^\prime(x)=\frac{1}{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)=f(x)2-2g(x)h(x) are pfaffian.

[edit] Rigorous Definition

Let U be an open domain in \mathbb{R}^n. A pfaffian chain of order r ≥ 0 and degree α ≥ 1 in U is a sequence of real analytic functions f1,…, fr 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}\in\mathbb{R}[x_1,\,\ldots,\,x_n,\,y_{1},\ldots,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\in\mathbb{R}[x_1,\,\ldots,\,x_n,\,y_{1},\,\ldots,y_{r}]\, is a polynomial of degree at most β ≥ 1.

[edit] Examples

  1. Any polynomial is a pfaffian function with r = 0.
  2. The function f(x)=ex 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.

[edit] References

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