Abstract
For a commutative ring A, a polynomial f ∈ A[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.
MSC : 14R10, 14R25
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Notes
- 1.
Some authors have used the term x-residual coordinate instead; however, as the definition does not depend on the choice of the variable x, we will stick with residual coordinate.
- 2.
Our construction provides a different coordinate system than Freudenburg’s.
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Acknowledgements
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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Lewis, D. (2014). Strongly Residual Coordinates over A[x]. In: Cheltsov, I., Ciliberto, C., Flenner, H., McKernan, J., Prokhorov, Y., Zaidenberg, M. (eds) Automorphisms in Birational and Affine Geometry. Springer Proceedings in Mathematics & Statistics, vol 79. Springer, Cham. https://doi.org/10.1007/978-3-319-05681-4_23
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DOI: https://doi.org/10.1007/978-3-319-05681-4_23
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