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The bounds of rigid sphere design

Yuhi Kamio

Abstract

We have identified some necessary conditions for the existence of rigid sphere designs. In particular, we have successfully resolved the conjecture proposed by [Ban87]; Given fixed positive integers t and d, we show that there exist only finitely many rigid t-designs on Sd, up to orthogonal transformations.

The bounds of rigid sphere design

Abstract

We have identified some necessary conditions for the existence of rigid sphere designs. In particular, we have successfully resolved the conjecture proposed by [Ban87]; Given fixed positive integers t and d, we show that there exist only finitely many rigid t-designs on Sd, up to orthogonal transformations.
Paper Structure (1 section, 2 theorems, 3 equations)

This paper contains 1 section, 2 theorems, 3 equations.

Table of Contents

  1. Acknowledgement

Key Result

Lemma 2

Let $f_1,f_2,\ldots ,f_s\in \mathbb{R}[x_1,\ldots ,x_n]$ be polynomials whose degrees are at most $t$. Then, the number of isolated common real roots of $f_1, f_2,\ldots ,f_s$ is at most $t(2t-1)^{n-1}$.

Theorems & Definitions (5)

  • Definition 1
  • Lemma 2
  • proof
  • Theorem 3
  • proof