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2 results

  • 14 - Algebraic Families, Dimension Counting, and Ruled Surfaces

  • Adam Sheffer, Bernard M. Baruch College, City University of New York
  • Book:
    Polynomial Methods and Incidence Theory
    Published online:
    17 March 2022
    Print publication:
    24 March 2022, pp 204-224

  • Summary

    In this chapter we discuss advanced tools and techniques, which rely on additional concepts from algebraic geometry. These tools could be helpful for people who do research work in incidence theory and related topics. A reader who is new to this field might prefer to skip this chapter.

    We sometimes wish to consider families of varieties, such as the set of circles in the plane or the set of planes in R^3 that not are incident to the origin. In this chapter, we rigorously define such families. We also generalize the idea of point-line duality to every family of varieties. We then see how these notions could be used to prove various results. In particular, we derive a new incidence bound and prove various properties of surfaces in R^3 and C^3.

  • ASYMPTOTICS OF A GAUSS HYPERGEOMETRIC FUNCTION WITH TWO LARGE PARAMETERS: A NEW CASE

  • Part of
  • J. F. HARPER
  • Journal:
    The ANZIAM Journal / Volume 62 / Issue 4 / October 2020
    Published online by Cambridge University Press:
    10 December 2019, pp. 446-452

  • Asymptotic expansions of the Gauss hypergeometric function with large parameters, $F(\unicode[STIX]{x1D6FC}+\unicode[STIX]{x1D716}_{1}\unicode[STIX]{x1D70F},\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D716}_{2}\unicode[STIX]{x1D70F};\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D716}_{3}\unicode[STIX]{x1D70F};z)$ as $|\unicode[STIX]{x1D70F}|\rightarrow \infty$, are known for many special cases, but not for one that the author encountered in recent work on fluid mechanics: $\unicode[STIX]{x1D716}_{2}=0$ and $\unicode[STIX]{x1D716}_{3}=\unicode[STIX]{x1D716}_{1}z$. This paper gives the leading term for that case if $\unicode[STIX]{x1D6FD}$ is not a negative integer and $z$ is not on the branch cut $[1,\infty )$, and it shows how subsequent terms can be found.