Suppose I have a set of random vectors $f(a\_1, \ldots, a\_\ell) := (v\_1, \ldots, v\_m) \subset F\_p^n$, $m \ge n$, given by a matrix valued polynomial function $f$, where the $a\_i$'s are independent, uniformly distributed in $F\_p \setminus \{0\}$, and each component of $f$ consists of some polynomial in the $a\_i$'s whose coefficients are all in $\{0,1\}$, and such that the degree of each variable is at most $1$. For example, the following set of random vectors fits the discription:


$$ v\_1 = (a\_1, a\_2 a\_1, a\_3); v\_2 = (a\_2, a\_1 a\_2 a\_3 + a\_2, 0); v\_3 = (a\_1 + a\_2, a\_2 + a\_3, a\_3 a\_1)$$.


Now consider the quantity $\pi\_p = \mathbb{P}(f(a\_1, \ldots, a\_\ell) \text{ spans } F\_p^n)$. My question is, is $\pi\_p$ monotone non-decreasing in $p$? If not can one give a counterexample? The motivation comes from a recent result of Yuval Peres and Allan Sly (Arxiv preprint arXiv:1105.4402, 201) giving the right order of mixing time for the most natural random walk on uni-upper triangular matrices over $F\_p$. Knowing the above will extend their result of $\mathcal{O}(n^2)$ to $p$ that grows with $n$, which is highly anticipated. 


Edit: Will Sawin below essentially solved an earlier version of this problem, where I forgot to state the degree condition on the $a\_i$'s.



Probability of a set of random vectors over finite field being a spanning set
One vector $v=a\_1^2+1$. The probability that it forms a spanning set is $1$ or less than $1$ depending on if $-1$ is a quadratic residue mod $p$.


Edit: In the linear case, you can just set $v\_1=(a\_1,a\_2)$, $v\_2=(a\_2,-a\_1)$, determinant of the matrix $=a\_1^2+a\_2^2$. They span if and only if the determinant is nonzero, which happens if and only if $(a\_1/a\_2)^2\neq-1$.


