Strongly Residual Coordinates over A[x]

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 79)


For a commutative ring A, a polynomial fA[x][n] is called a strongly residual coordinate if f becomes a coordinate (over A) upon going modulo x, and f becomes a coordinate (over A[x, x−1]) upon inverting x. We study the question of when a strongly residual coordinate in A[x][n] is a coordinate, a question closely related to the Dolgachev–Weisfeiler conjecture. It is known that all strongly residual coordinates are coordinates for n = 2 over an integral domain of characteristic zero. We show that a large class of strongly residual coordinates that are generated by elementaries over A[x, x−1] are in fact coordinates for arbitrary n, with a stronger result in the n = 3 case. As an application, we show that all Vénéreau-type polynomials are 1-stable coordinates.



The author would like to thank David Wright, Brady Rocks, and Eric Edo for helpful discussions and feedback. The author is also indebted to the referee for several helpful suggestions.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.University of AlabamaTuscaloosaUSA

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