Computing the Elementary Symmetric Polynomials in Positive Characteristics

We first extend the results of Chatterjee,Kumar,Shi,Volk(Computational Complexity 2022) by showing that the degree $d$ elementary symmetric polynomials in $n$ variables have formula lower bounds of $Ω(d(n-d))$ over fields of positive characteristic. Then, we show that the results of the universality of linear projections of elementary symmetric polynomials from Shpilka(JCSS 2002) and of border fan-in two $ΣΠΣ$ circuits from Kumar(ACM TOCT 2020) over zero characteristic fields do not extend to fields of positive characteristic. In particular, we show that *There are polynomials that cannot be represented as linear projections of the elementary symmetric polynomials(in fact, we show linear lower bounds over the size of the sum of such linear projections) and *There are polynomials that cannot be computed by border depth-$3$ circuits of top fan-in $k$, called $\overline{Σ^{[k]}ΠΣ}$, for $k = o(n)$. To prove the first result, we consider a geometric property of the elementary symmetric polynomials, namely, the set of all points in which the polynomial and all of its first-order partial derivatives vanish. It was previously shown that the dimension of this space was exactly $d-2$ for fields of zero characteristic. We extend this to fields of positive characteristic by showing that this dimension must be between $d-2$ and $d-1$. In fact, we provide some criterion where it is $d-2$ and others where it is $d-1$. Then, to consider the border top fan-in of linear projections of the elementary symmetric polynomials and border depth-$3$ circuits(sometimes called border affine Chow rank), we show that it is sufficient to consider the border top fan-in of the sum of such linear projections of the elementary symmetric polynomials. This is done by an explicit construction of a ‘metapolynomial,’ meaning that this result also applies in the border setting.

💡 Research Summary

This paper studies the computational complexity of the elementary symmetric polynomial eₙ,₍d₎ in fields of positive characteristic and shows that several powerful results known over characteristic‑zero fields fail to extend. The authors obtain three main contributions.

1. Formula lower bounds in positive characteristic.
Building on the recent work of Chatterjee, Kumar, Shi and Volk (2022), the authors prove that any algebraic formula computing eₙ,₍d₎ must have size Ω(d·(n−d)) even when the underlying field F has char(F)=p>0. The proof hinges on the geometry of the “order‑2 zero space”
V₂(f) = { a ∈ Fⁿ | f(a)=∂f/∂x₁(a)=…=∂f/∂xₙ(a)=0 }.
For characteristic zero it is known that dim V₂(eₙ,₍d₎)=d−2. The authors show that in positive characteristic the dimension can be either d−2 or d−1, depending on arithmetic relations between n, d and the characteristic p. Lemma 1.7 gives a precise criterion: if p≠0, n−d+1≡0 (mod p⌈logₚ d⌉) and n≥2d−1, then dim V₂(eₙ,₍d₎)=d−1; otherwise dim V₂(eₙ,₍d₎)=d−2. Since the dimension never exceeds d−1, Lemma 1.5 (which relates dim V₂ to the number k of product terms in a decomposition f=∑_{i=1}^{k} f_i g_i + h) yields the inequality n−2k ≤ dim V₂ ≤ d−1. Rearranging gives k ≥ Ω(d·(n−d)), establishing the desired lower bound. This matches the well‑known O(n²) upper bound from the Ben‑Or ΣΠΣ construction, making the bound tight for many parameter regimes.

2. Failure of universality of linear projections.
In characteristic zero, Shpilka (2002) proved that every polynomial can be expressed as a linear combination of elementary symmetric polynomials after applying linear projections (the “symmetric model” is universal). The authors refute this universality for fields with char(F)≠0. Theorem 1.2 states that for any n there exists a homogeneous degree‑d polynomial f such that for any collection of linear forms L^{(i)}₁,…,L^{(i)}{m_i} and scalars c_i, with k=o(n), we have
f ≠ ∑{i=1}^{k} c_i e_d(L^{(i)}₁,…,L^{(i)}{m_i}).
Moreover, the impossibility persists in the border (ε‑approximation) setting. The proof introduces a “metapolynomial” M(x,α)=∑{i=1}^{k} α_i e_d(L_i(x)), where the α_i are new indeterminates. This construction collapses the sum of many projected elementary symmetric polynomials into a single polynomial whose order‑2 zero space can be analyzed. If k were sublinear, the resulting V₂(M) would be too large (≥ n−O(k)), contradicting the bound dim V₂(eₙ,₍d₎)≤d−1 from Lemma 1.7. Hence no sublinear number of projections suffices, and the symmetric model is not universal in positive characteristic.

**3. Border Σ^{