# Subspaces of vanishing permanent

Suppose that $$p\ge 5$$ is a prime, $$n$$ a positive integer divisible by $$p-1$$, and $$L<\mathbb F_p^n$$ a subspace of dimension $$d=n/(p-1)$$. Do there exist vectors $$l_1,\dotsc,l_n\in L$$ such that the matrix with $$l_1,\dotsc,l_n$$ as its columns has a nonzero permanent? Clearly, the answer is negative if $$L$$ is contained in a coordinate hyperplane, or in a linear subspace like $$\{(x_1,\dotsc,x_n)\in\mathbb F_p^n\colon x_1=\dotsb=x_p\}$$; are there other obstructions of this sort?

Is it possible to classify those subspaces $$L<\mathbb F_p^n$$ for which any square matrix of order $$n$$ with all its column vectors in $$L$$ has a vanishing permanent?

Notice that if the permanent vanishes for $$l_1,\dotsc,l_n$$ being the elements of some particular basis of $$L$$, with each element repeated $$p-1$$ times, then in fact it vanishes for any $$l_1,\dotsc, l_n\in L$$. (It is this property that depends critically on the assumption $$d=n/(p-1)$$.)

In the case where $$d=1$$, the requirement that $$L$$ is not contained in a coordinate hyperplane is easily seen to be also sufficient. The case $$d=2$$ does not look that easy to me.

• Your highlighted question doesn't include the hypothesis on the dimension of $L$ that is in the introductory paragraph. Do you want to continue imposing that hypothesis? Commented Apr 20, 2021 at 18:43
• @LSpice: Absolutely; everything is about subspaces $L$ of dimension $d=n/(p-1)$.
– Seva
Commented Apr 20, 2021 at 18:56

Let $$v_1,\dots, v_d$$ be a basis for $$L$$. Then the permanent of the matrix obtained from $$p-1$$ repetitions each of $$v_1,\dots, v_d$$ is a polynomial function in the entries of $$v_1,\dots, v_d$$. Keeping all the entries but the last one in each $$v_i$$ constant, we get a linear function. Since it is linear, among all $$p^{nd}$$ possible tuples of vectors, the number that satisfy it is at least $$p^{nd-1}$$.
Not all $$d$$-tuples of vectors in $$\mathbb F_p^n$$ form a basis, but $$1 - O( p^{d-n-1})$$ do.
So the fraction of subspaces $$L$$ where this determinant vanishes is at least $$\frac{1}{p} - O( p^{d-n-1})$$
On the other hand, the fraction that satisfy one of your conditions is smaller than that for $$d>1$$ - just $$\frac{ d (p-1)}{p^d}$$ satisfy the coordinate condition and $$\frac{ \binom{d (p-1) }{ p } } { p^{ (p-1) d }}$$ satisfy the $$p$$-fold repetition condition.