In this paper, we study the computational complexity of the commutative determinant polynomial computed by a class of set-multilinear circuits which we call regular set-multilinear circuits. Regular set-multilinear circuits are commutative circuits with a restriction on the order in which they can compute polynomials. A regular circuit can be seen as the commutative analogue of the ordered circuit defined by Hrubes,Wigderson and Yehudayoff [HWY10]. We show that if the commutative determinant polynomial has small representation in the sum of constantly many regular set-multilinear circuits, then the commutative permanent polynomial also has a small arithmetic circuit.

翻译：在本文中,我们研究了由一组固定的多线性电路(我们称之为固定的多线性电路)计算成的交点决定因素多线性多线性电路的计算复杂性。固定的固定的多线性电路是流动性电路,限制他们计算多线性电路的顺序。正常的电路可以被视为由Hrubes、Wigderson和Yehudayoff[HWY10]定义的定序电路的交点性模拟物。我们表明,如果交点决定性多线性电路在经常许多固定的固定多线性电路的总和中代表较少,那么循环性永久多线性电路也有一个小的计算线路。