This was something cool I saw during Error Correcting Codes. Let $\gamma$ be a primitive root mod $p$, and let $f,g$ be two polynomials of degree at most $p-2$ mod $p$.

Suppose that $f,g$ vanish on $\gamma,\gamma^2, \cdots, \gamma^{p-k-1}$. Show that the coefficientwise product $f\odot g$ vanishes on $\gamma,\cdots, \gamma^{p-2k}$, defined by $$(f\odot g)(x) = \sum_{i=0}^{p-1} f_ig_i x^i$$ if $f(x) = \sum_i f_ix^i, g(x) = \sum_i g_ix^i$.

Left on the cutting floor from my probability notes.

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Context

This technique is best appreciated when we’ve understood the two phenomena below:

(1) Randomly picking an object is decent on average but performs badly near the “tails”:

Problem. (The stamp collector problem) McDonald’s has decided to release a series of $n$ different Pokemon Happy Meal toys.

1. Show that the expected number of Happy Meals required to collect 90% of all the toys is around $n\ln 10$.
2. But if we truly ``gotta catch ’em all’’, show that the expected number of Happy Meals required is instead around $n\ln n$.

For large $n$, these are very different!

This is a “Power Round”-style problem set exploring how facts in Euclidean geometry can be “extended” to complex numbers.

This was something I learnt from Prof. Umut Varoglunes during the graduate differential topology class (Math 215B) at Stanford.

Suppose you had a differentiable map $(a,b) \to \mathbb R$. If it is locally injective, must it be (globally) injective?