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.

Definition

Let $U\subset \mathbb {R} ^{n}$ be an open domain. A pfaffian chain of order $r\geq 0$ and degree $\alpha \geq 1$ in ${U}$ is a sequence of real analytic functions $\,f_{1},\,\ldots ,\,f_{r}\,$ 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 $i=1,\ldots r$ where $P_{i,j}\in [x_{1},\,\ldots ,\,x_{n},\,y_{1},\ldots ,y_{i}]$ are polynomials of degree $\leq \alpha$. A function $f$ on $U$ is called a pfaffian function of order $r$ and degree $(\alpha ,\beta )$ if

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

where $\,P\in [x_{1},\,\ldots ,\,x_{n},\,y_{1},\,\ldots ,y_{r}]\,$ is a polynomial of degree at most $\beta \geq 1$.

Examples

1. Any polynomial is a pfaffian function with $r=0$.
2. The function $f(x)=e^{x}$ is pfaffian with $r=1$ and $\alpha =\beta =1$ due to the equation $f^{\prime }=f$.
3. The algebraic functions are pfaffian.
4. Any combination of polynomials, exponentials, the trigonomtric functions on bounded intervals, and their inverses, in any finite number of variables, is pfaffian.

References

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