TAOCP 4.6.2 Exercise 1
The proposed solution does not answer the exercise that was asked.
Section 4.6.2: Factorization of Polynomials
Exercise 1. ▶ [M25] Let $p$ be prime, and let $u(x)$ be a random polynomial of degree $n$, assuming that each of the $p^n$ monic polynomials is equally likely. Show that if $n \ge 2$, the probability that $u(x)$ has a linear factor mod $p$ is approximately $(1 - (1 - 1/p)^p)/3$, inclusive. Give a closed form for this probability when $n \ge p$. What is the average number of linear factors?
Verified: no
Solve time: 4m39s
Correctness
The proposed solution does not answer the exercise that was asked.
Exercise 4.6.1.23 concerns the sequence of polynomials $u_j(x)$ appearing in equation (29) of Section 4.6.1, and asks for a proof that if $u(x)$ has $\deg(u)$ real roots, then
$$ \deg(u_{j+1})=\deg(u_j)-1,\qquad 0\le j\le k. $$
A correct solution must use the definitions and properties of the polynomial remainder sequence in (29), together with the hypothesis that all roots of $u(x)$ are real.
The proposed solution instead discusses approximate polynomial gcds, floating-point arithmetic, numerical instability of Euclid's algorithm, Sylvester matrices, and approximate common divisors. None of these topics addresses the stated theorem. No argument involving the polynomials $u_j(x)$ is given, and equation (29) is never mentioned.
Therefore the solution does not prove the required result.
Gaps and Errors
Critical error. The solution solves a different problem. It discusses approximate polynomial gcds and floating-point coefficients, while the exercise asks for a proof about degree drops in the sequence $u_j(x)$ under the hypothesis that $u(x)$ has only real roots.
Critical error. No use is made of equation (29), the sequence $u_j$, or the hypothesis that $u(x)$ has $\deg(u)$ real roots.
Critical error. The conclusion
$$ \deg(u_{j+1})=\deg(u_j)-1 $$
is never established.
Because the submitted work is unrelated to the stated exercise, there is no mathematical proof of the required claim to evaluate.
Summary
The solution is entirely off-topic. It contains a reasonable discussion of approximate polynomial gcds, but it does not address Exercise 4.6.1.23 and provides no proof of the stated theorem.
VERDICT: FAIL, the solution addresses approximate polynomial gcds instead of proving the degree relation for the sequence $u_j$ in equation (29).