The Constant Degree Hypothesis was introduced by Barrington et. al. (1990) to study some extensions of $q$-groups by nilpotent groups and the power of these groups in a certain computational model. In its simplest formulation, it establishes exponential lower bounds for $\mathrm{AND}_d \circ \mathrm{MOD}_m \circ \mathrm{MOD}_q$ circuits computing AND of unbounded arity $n$ (for constant integers $d,m$ and a prime $q$). While it has been proved in some special cases (including $d=1$), it remains wide open in its general form for over 30 years. In this paper we prove that the hypothesis holds when we restrict our attention to symmetric circuits with $m$ being a prime. While we build upon techniques by Grolmusz and Tardos (2000), we have to prove a new symmetric version of their Degree Decreasing Lemma and apply it in a highly non-trivial way. Moreover, to establish the result we perform a careful analysis of automorphism groups of $\mathrm{AND} \circ \mathrm{MOD}_m$ subcircuits and study the periodic behaviour of the computed functions. Finally, our methods also yield lower bounds when $d$ is treated as a function of $n$.

翻译：常度数假设由Barrington等人（1990）提出，用于研究$q$-群通过幂零群的某些扩展以及这些群在特定计算模型中的能力。在最简形式下，该假设为计算无界元数$n$的AND函数的$\mathrm{AND}_d \circ \mathrm{MOD}_m \circ \mathrm{MOD}_q$电路（其中$d,m$为常数整数，$q$为素数）建立了指数级下界。尽管该假设已在某些特殊情形（包括$d=1$）下得到证明，但其一般形式在三十余年来仍悬而未决。本文证明，当我们将注意力限制在$m$为素数的对称电路时，该假设成立。虽然我们借鉴了Grolmusz和Tardos（2000）的技术，但必须证明其“度递减引理”的一个新对称版本，并以高度非平凡的方式应用该引理。此外，为确立此结果，我们对$\mathrm{AND} \circ \mathrm{MOD}_m$子电路的群自同构进行了精细分析，并研究了被计算函数的周期性行为。最后，我们的方法还在将$d$视为$n$的函数时得出了下界结果。