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Generic polynomial
About: Generic polynomial is a research topic. Over the lifetime, 608 publications have been published within this topic receiving 6784 citations.
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TL;DR: In this paper, the authors define the notion of nilpotency of elements in a polynomial algebra over a field of characteristic not 2 or 3 and prove that the algebra of polynomials in y over y is associative.
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TL;DR: In this article, a generic Galois-extension of a finite-dimensional algebra over a finite field was constructed explicitly, where $R$ is a localized polynomial ring over the field.
Abstract: Let $\mathcal{A}$ be a finite-dimensional algebra over a finite field $\mathbf{F}_q$ and let $G=\mathcal{A}^\times$ be the multiplicative group of $\mathcal{A}$. In this paper, we construct explicitly a generic Galois $G$-extension $S/R$, where $R$ is a localized polynomial ring over $\mathbf{F}_q$, and an explicit generic polynomial for $G$ in $\dim_{\mathbf{F}_q}(\mathcal{A})$ parameters.
TL;DR: In this paper, the field isomorphism problem of cubic generic polynomial X3+sX+s over the field of rational numbers with the specialization of the parameter s to nonzero rational integers m via primitive solutions to the family of cubic Thue equations x3−2mx2y−9mxy2−m(2m+27)y3=λ.
Abstract: We study the field isomorphism problem of cubic generic polynomial X3+sX+s over the field of rational numbers with the specialization of the parameter s to nonzero rational integers m via primitive solutions to the family of cubic Thue equations x3−2mx2y−9mxy2−m(2m+27)y3=λ where λ2 is a divisor of m3(4m+27)5.
01 Jan 1991
TL;DR: In this article, it was shown that the polynomial X does not have controlled singularities so this case can indeed be discarded in the proof of Proposition 5, and a proof for the existence of identically zero polynomials is given.
Abstract: The proof of Proposition 5 of [3] is incomplete. With notation as in the paper, the possibility that the polynomial X~/q'P o + X~/~' P1 + X~/q" Pz in (11) could be identically zero was overlooked. We will sketch here a proof that in this case X does not have controlled singularities so this case can indeed be discarded in the proof of Proposition 5. 2 Let F = Z SIPS" be a generic polynomial of this form with degPi = )~, (so i=0 2 d = degF = 2q' + 1) with ~2 X~/q'Pi identically zero. Thus every common zero of Po /=0 and P1 is a zero of X~/q' P2 and, since we are in the generic case, is a zero of PE and gives a singular point of F = 0 with Jacobian ideal of multiplicity at least q' since c~F/~X i = P~'.
TL;DR: In this paper, the authors studied the differential field extensions of F that satisfy the first condition but not the second condition, that is, their field of constants coincides with the algebraically closed field of constant constants.