We study algebraic complexity classes and their complete polynomials under \emph{homogeneous linear} projections, not just under the usual affine linear projections that were originally introduced by Valiant in 1979. These reductions are weaker yet more natural from a geometric complexity theory (GCT) standpoint, because the corresponding orbit closure formulations do not require the padding of polynomials. We give the \emph{first} complete polynomials for VF, the class of sequences of polynomials that admit small algebraic formulas, under homogeneous linear projections: The sum of the entries of the non-commutative elementary symmetric polynomial in 3 by 3 matrices of homogeneous linear forms. Even simpler variants of the elementary symmetric polynomial are hard for the topological closure of a large subclass of VF: the sum of the entries of the non-commutative elementary symmetric polynomial in 2 by 2 matrices of homogeneous linear forms, and homogeneous variants of the continuant polynomial (Bringmann, Ikenmeyer, Zuiddam, JACM '18). This requires a careful study of circuits with arity-3 product gates.

翻译：我们研究在齐次线性投影下的代数复杂性类及其完全多项式，而不仅限于Valiant在1979年最初引入的通常仿射线性投影。从几何复杂理论（GCT）的角度看，这些约化虽然较弱，但更为自然，因为相应的轨道闭包表述不需要对多项式进行填充。我们给出了VF（即允许小代数公式的多项式序列类）在齐次线性投影下的首个完全多项式：由齐次线性形式构成的3×3矩阵的非交换初等对称多项式的条目和。甚至更简单的初等对称多项式变种对于VF的某个大子类的拓扑闭包也是困难的：由齐次线性形式构成的2×2矩阵的非交换初等对称多项式的条目和，以及连续多项式（Bringmann, Ikenmeyer, Zuiddam, JACM '18）的齐次变种。这需要对带有3元乘积门的电路进行仔细研究。