# The eliminant of a system of differential equations

I am reading an old paper dealing with linear differential operators. At one point it refers to something it calls the "eliminant" of a set of linear differential operators. It seems that this was a well-known concept at the time (the 1920-ies) but I have not heard of it before.

I think I can guess what the eliminant is, at least in the particular case I the paper considers, from context. But I would really need to read up on the theory for eliminants to fully understand the paper. Does anyone have a reference?

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Could it be the resultant? Superficial googling seems to suggest that. – Thorny Jan 6 '10 at 12:09
Thorny is right. It's the resultant. This is the first google hit for "eliminant": mathworld.wolfram.com/Eliminant.html . – José Figueroa-O'Farrill Jan 6 '10 at 15:38
The eliminant or resultant as given there us defined for polynomials while I am interested in the case of differential operators unfortunately. – Johan Jan 7 '10 at 18:41
Then it's probably referring to the resultant of their symbols, that is, replace the operator $\sum f_i(x)D^i$ by $\sum f_i(x)y^i$, and take the resultant in the $y$ variable. – Charles Siegel Jan 11 '10 at 1:19

In "Leopold Kroneckers Werke, 3. Band", Teubner (1899), published by K. Hensel, we find on p. 179 in a section about applications of modulsystems the phrase: "Die Resultante der Elimination von ...".

This suggests that Thorny's above conjecture is correct and the eliminant is also called resultant. The due reference given by José Figueroa-O'Farrill is confirmed by the McGraw-Hill Dictionary of Scientific & Technical Terms.

The eliminant has originally been defined for algebraic equations. See Otto Biermann's comprehensive and detailed paper: "Über die Bildung der Eliminanten eines Systems algebraischer Gleichungen", Monatshefte für Mathematik und Physik (1894) pp. 17-32, referring (without giving sources) to Salmon-Fiedler, Günther, Sylvester and Cayley. The mode of application to linear differential operators has been suggested here by Charles Siegel, and application to linear differential equations becomes obvious when the characteristic polynoms are formed in the common way by means of the exponential function.

Application of eliminant to differential equations can be found in Paul Funk's paper "Beiträge zur zweidimensionalen Finsler'schen Geometrie", Monatshefte für Mathematik 52 (1948) pp. 194-216, and also in modern literature, namely in A.P. Alexandrov's arXiv-paper: "Dynamic systems with quantum behaviour" on p. 99.

Apropos The word "eliminante" has acquired a general use in mathematics. This can be seen by the completely independent application of the word by Hermann Weyl in his paper "Reine Infinitesimalgeometrie" Mathematische Zeitschrift 2 (1918) pp 384-411.

Jene fünf Identitäten stehen in engstem Zusammenhang mit den sog. Erhaltungssätzen, nämlich dem (einkomponentigen) Satz yon der Erhalung der Elektrizität und dem (vierkomponentigen) Energie-Impulsprinzip. Sie lehren nämlich: die Erhaltungssätze (auf deren Gültigkeit die Mechanik beruht) folgen auf doppelte Weise aus den elektromagnetischen sowie den Gravitationsgleichungen; man möchte sie daher als die gemeinsame Eliminante dieser beiden Gesetzesgruppen bezeichnen.

Briefly: The conservation principles follow from the electromagnetic and the gravitational equations; one is tempted to denote them as common eliminant of these two groups of laws.

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