Timeline for Factoring $x^p H(x) + x^q B(x) + T(x)$ over a finite field

6 events

when toggle format	what		by	license	comment
May 4, 2019 at 13:53	history	edited	Seva	CC BY-SA 4.0	added 218 characters in body; edited title
May 4, 2019 at 6:53	comment	added	alpoge		Ah! I tacitly assumed the discriminant was nonzero when I saw split completely (I’m not sure why). Alas!
May 4, 2019 at 6:30	comment	added	Seva		@alpoge: Replacing $x^p$ with $x$, you change the polynomial. The new polynomial will have the same roots as the original one, but the multiplicities can be affected, and full reducibility is not necessarily preserved (consider $x^{p-1}(x-1)$).
May 3, 2019 at 21:32	comment	added	alpoge		Sorry, I am probably confused and if so forgive me for what is likely a dumb remark, but for x\in \F_p you may replace the x^p term by simply x, and then you are asking that x^q B(x) + (x H(x) + T(x)) have q + \deg B roots mod p, i.e. you are asking for conditions on the leading \deg B (< 0.1 (p-1) ) coefficients of a polynomial over \F_p (of degree < p) of the form const * \prod_{\alpha\in S} (x - \alpha), where S\subseteq \F_p. [This recovers the 2^p vs. p^{0.1 p} comparison, of course.] It would be particularly nice if \deg B = p-1-q, in which case up to translation you’d have x^{p-1} - 1.
May 3, 2019 at 13:50	answer	added	Fedor Petrov		timeline score: 3
May 3, 2019 at 13:22	history	asked	Seva	CC BY-SA 4.0