Celebratio Mathematica

W. D. Neumann and P. Norbury: “Nontrivial rational polynomials in two variables have reducible fibres,” Bull. Aust. Math. Soc. 58 : 3 (1998), pp. 501–503. MR 1662136 Zbl 0946.14038 ArXiv math/9805093 article

W. D. Neumann: “Notes on geometry and 3-manifolds,” pp. 191–267 in Low dimensional topology (Eger, Hungary, 29 July–2 August 1996 and Budapest, 3–14 August 1998). Edited by K. Böröczky, Jr., W. Neumann, and A. Stipsicz. Bolyai Society Mathematical Studies 8. János Bolyai Mathematical Society (Budapest), 1999. With appendices by Paul Norbury. MR 1747270 Zbl 0944.57012 incollection

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W. D. Neumann and P. Norbury: “Vanishing cycles and monodromy of complex polynomials,” Duke Math. J. 101 : 3 (2000), pp. 487–497. MR 1740685 Zbl 0978.32030 article

In this paper we describe the trivial summand for monodromy around a fibre of a polynomial map \( \mathbb{C}^n\to\mathbb{C} \), generalising and clarifying work of Artal Bartolo, Cassou-Noguès and Dimca [1998], who proved similar results under strong restrictions on the homology of the general fibre and singularities of the other fibres. They also showed a polynomial map \( f:\mathbb{C}^2\to \mathbb{C} \) has trivial global monodromy if and only if it is “rational of simple type” in the terminology of Miyanishi and Sugie. We refine this result and correct the Miyanishi–Sugie classification of such polynomials, pointing out that there are also non-isotrivial examples.

W. D. Neumann and P. Norbury: “Unfolding polynomial maps at infinity,” Math. Ann. 318 : 1 (2000), pp. 149–180. MR 1785580 Zbl 1005.32021 ArXiv math/9910054 article

W. D. Neumann and P. Norbury: “Rational polynomials of simple type,” Pac. J. Math. 204 : 1 (2002), pp. 177–207. MR 1905197 Zbl 1055.14062 ArXiv math/0008080 article

W. D. Neumann and P. Norbury: “The Orevkov invariant of an affine plane curve,” Trans. Am. Math. Soc. 355 : 2 (2003), pp. 519–538. MR 1932711 Zbl 1056.14084 ArXiv math/0110286 article

J. H. Manton, W. D. Neumann, and P. T. Norbury: “On the algebraic identifiability of finite impulse response channels driven by linearly precoded signals,” Systems Control Lett. 54 : 2 (2005), pp. 125–134. MR 2109579 Zbl 1129.93358 article

It is common in wireless communications to perform some form of linear precoding operation on the source signal prior to transmission over a channel. Although the traditional reason for precoding is to improve the performance of the communication system, this paper draws attention to the fact that the receiver can identify the impulse response of the channel without any prior knowledge of the transmitted signal simply by solving a system of polynomial equations. Since different precoders lead to different systems of equations, this paper addresses the fundamental issue of determining which classes of linear precoders lead to a system of equations having a unique solution. In doing so, basic properties of polynomial equations which are useful for studying other identifiability issues commonly encountered in engineering and the applied sciences are presented.

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