In mathematical analysis, the characteristic variety of a

microdifferential operator In mathematics, a differential operator is an operator defined as a function of the differentiation operator. It is helpful, as a matter of notation first, to consider differentiation as an abstract operation that accepts a function and retur ...

''P'' is an

algebraic variety Algebraic varieties are the central objects of study in algebraic geometry, a sub-field of mathematics. Classically, an algebraic variety is defined as the set of solutions of a system of polynomial equations over the real or complex numbers. Mo ...

that is the zero set of the

principal symbol In mathematics, the symbol of a linear differential operator is a polynomial representing a differential operator, which is obtained, roughly speaking, by replacing each partial derivative by a new variable. The symbol of a differential operat ...

of ''P'' in the

cotangent bundle In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...

. It is invariant under a

quantized contact transformation Quantization is the process of constraining an input from a continuous or otherwise large set of values (such as the real numbers) to a discrete set (such as the integers). The term ''quantization'' may refer to: Signal processing * Quantizatio ...

. The notion is also defined more generally in

commutative algebra Commutative algebra, first known as ideal theory, is the branch of algebra that studies commutative rings, their ideals, and modules over such rings. Both algebraic geometry and algebraic number theory build on commutative algebra. Prominent ...

. A basic theorem says a characteristic variety is involutive.

References

* M. Sato, T. Kawai, and M. Kashiwara: Microfunctions and Pseudo-differential Equations. Lecture note in Math., No. 287, Springer, Berlin-Heidelberg-New York, pp. 265–529 (1973) {{analysis-stub Algebraic varieties Mathematical analysis