Celebratio Mathematica — Doob — Complete Bibliography

Complete Bibliography

Works connected to Joseph Fels Ritt

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J. F. Ritt and J. L. Doob: “Systems of algebraic difference equations,” Bull. Am. Math. Soc. 39 : 7 (1933), pp. 508. JFM 59.0461.04 article

J. F. Ritt and J. L. Doob: “Systems of algebraic difference equations,” Am. J. Math. 55 : 1–4 (1933), pp. 4. A research announcement for this was published in Bull. Am. Math. Soc. 39:7 (1933). MR 1506981 JFM 59.0456.01 Zbl 0008.01804 article

The object of this paper is to derive an analogue, for systems of algebraic difference equations, of the fundamental theorem in the theory of systems of algebraic differential equations developed by [Ritt 1932]. We introduce the notion of irreducible system of algebraic difference equations, and show that every system of such eqiiations is equivalent to a finite set of irreducible systems. It will possibly strike one as curious that so general a result should be obtainable at a time when existence theorems for non-linear difference equations are almost entirely lacking.

Although our proof resembles greatly that for differential equations, there are also essential differences. These arise out of the circumstance that the derivative of a polynomial in several functions involves the derivatives of the functions linearly, while no corresponding result holds for the operation of differencing.