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Register a new userAnders Wimans On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations
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Alexander J. Sutherland
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This is an English translation of Anders Wimans classical paper {U}ber die Anwendung der Tschirnhausen Transformation auf die Reduktion algebraischer Gleichungen from 1927. The original work first appeared in the 1927 extraordinary edition of Nova Acta Regiae Societatis Scientiarum Upsaliensis. In this paper, Wiman gives an argument that the general polynomial of degree nine can be solved using one algebraic function of four variables and accessory irrationalities of degree at most five. However, his argument assumes certain intersections in affine space are generic without proof.
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Let RD(n) denote the minimum d for which there exists a formula for the roots of the general degree n polynomial using only algebraic functions of d or fewer variables. In 1927, Hilbert sketched how the 27 lines on a cubic surface could be used to construct a 4-variable formula for the general degree 9 polynomial (implying $RD(9)le 4$). In this paper, we turn Hilberts sketch into a general method. We show this method produces best-to-date upper bounds on RD(n) for all n, improving earlier results of Hamilton, Sylvester, Segre and Brauer.
We translate Erland Samuel Brings treatise Meletemata quaedam Mathematica circa Transformationem Aequationum Alebraicarum (Some selected mathematics on the Transformation of Algebraic Equations) written as his Promotionschrift at the University of Lund in 1786, from its Latin into English, with modern mathematical notation. Bring (1736 - 98) made important contributions to algebraic equations and obtained the canonical form x^5+px+q = 0 for quintics before Jerrard, Ruffini and Abel. In due course, he realized the significance of the projective curve which now bears his name: the complete intersection of the homogeneous Fermat polynomials of degrees 1,2,3 in CP^4.
This is an English translation of G.N. Chebotarevs classical paper On the Problem of Resolvents, which was originally written in Russian and published in Vol. 114, No. 2 of the Scientific Proceedings of the V.I. Ulyanov-Lenin Kazan State University. In this paper, Chebotarev extends the method in Wimans On the Application of Tschirnhaus Transformations to the Reduction of Algebraic Equations to argue that the general polynomial of degree 21 admits a solution using algebraic functions of at most 15 variables. However, his and Wimans proofs assume that certain intersections in affine space are generic without proof.
We present two examples of reductions from the evolution equations describing discrete Schlesinger transformations of Fuchsian systems to difference Painleve equations: difference Painleve equation d-$Pleft({A}_{2}^{(1)*}right)$ with the symmetry group ${E}^{(1)}_{6}$ and difference Painleve equation d-$Pleft({A}_{1}^{(1)*}right)$ with the symmetry group ${E}^{(1)}_{7}$. In both cases we describe in detail how to compute their Okamoto space of the initial conditions and emphasize the role played by geometry in helping us to understand the structure of the reduction, a choice of a good coordinate system describing the equation, and how to compare it with other instances of equations of the same type.
This is an English translation of Felix Kleins paper Ueber die Transformation elfter Ordnung der elliptischen Functionen from 1879.
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