1

Assume that $\hat{U}$ and $\hat{V}$ are vectors of the form $\left[\alpha,\beta,0\right]$ and $\left[\gamma, \delta,\epsilon \right]$ in $R^3$ where $\alpha$, $\beta$, $\gamma$, $\delta$, and $\epsilon$ are all algebraic integers. We also assume that characteristic polynomial for $\alpha$ is degree 9 and $\beta$ is degree 12. Is it true that $\gamma \; \delta \; \epsilon$ all are degree 108 (9 $\cdot$ 12) or less, if it is known that $\hat{U}\bullet\hat{V}$ is degree 12?

I also know that degree of $\gamma$ is 12. How can I avoid a brute force search for $\delta$, $\epsilon$ ? Trying the LLL factorization search method takes too long for degrees > 72 if all are checked.

Thanks!

Randall
  • 91
  • Huh? How can you possibly conclude anything about the degree of $\epsilon$ when $U \cdot V$ has nothing to do with $\epsilon$? Are these assumed to be unit vectors? – Erick Wong Mar 20 '13 at 06:32

0 Answers0