We observe that the Kronecker product of tensors is the operation that converts the determinant polynomial into Cayley's first hyperdeterminant. We apply the Kronecker product to iterated matrix multiplication, which results in the hypercomputant, a VNP-complete and VW[1]-complete polynomial whose hardness we prove via the equivariance of the Kronecker product. The construction works over arbitrary commutative semirings and also for the tensor algebra and the exterior algebra. For the tensor algebra this gives a version of "noncommutative VNP", and for polynomials over the nonnegative real numbers this gives a version of "monotone VNP", each with the hypercomputant as the complete object. We take a parameterized complexity viewpoint and compare the noncommutative setting and the monotone setting. Using standard techniques we obtain optimal algebraic branching program width lower bounds in both settings, and these are notably not always the same. We also prove the polystability of the hypercomputant and that its isotypic components are characterized by their stabilizer.

翻译：我们观察到张量的Kronecker积是将行列式多项式转化为Cayley第一超行列式的运算。我们将Kronecker积应用于迭代矩阵乘法，从而得到超计算子（hypercomputant），这是一个VNP-完备且VW[1]-完备的多项式，其难度通过Kronecker积的等变性得以证明。该构造适用于任意交换半环，同时也适用于张量代数和外代数。对于张量代数，这给出了"非交换VNP"的一个版本；对于非负实数域上的多项式，这给出了"单调VNP"的一个版本，其中超计算子均为完备对象。我们从参数化复杂度视角出发，比较了非交换设定与单调设定的差异。利用标准技术，我们在两种设定下均获得了最优代数分支程序宽度下界，值得注意的是这些下界并不总是一致。我们还证明了超计算子的多稳定性，并指出其等型分量由稳定化子唯一刻画。