We study the annealed complexity of a random Gaussian homogeneous polynomial on the $N$-dimensional unit sphere in the presence of deterministic polynomials that depend on fixed unit vectors and external parameters. In particular, we establish variational formulas for the exponential asymptotics of the average number of total critical points and of local maxima. This is obtained through the Kac-Rice formula and the determinant asymptotics of a finite-rank perturbation of a Gaussian Wigner matrix. More precisely, the determinant analysis is based on recent advances on finite-rank spherical integrals by [Guionnet, Husson 2022] to study the large deviations of multi-rank spiked Gaussian Wigner matrices. The analysis of the variational problem identifies a topological phase transition. There is an exact threshold for the external parameters such that, once exceeded, the complexity function vanishes into new regions in which the critical points are close to the given vectors. Interestingly, these regions also include those where critical points are close to multiple vectors.

翻译：我们研究了在确定性多项式（依赖于固定单位向量和外部参数）存在时，$N$维单位球面上随机高斯齐次多项式的退火复杂度。具体而言，我们建立了总临界点和局部极大值平均数量的指数渐近变分公式。该结果通过Kac-Rice公式以及高斯Wigner矩阵有限秩扰动的行列式渐近分析得出。更精确地，行列式分析基于[Guionnet, Husson 2022]关于有限秩球面积分的最新进展，用于研究多秩尖峰高斯Wigner矩阵的大偏差。变分问题的分析揭示了一个拓扑相变：存在一个外部参数的精确阈值，一旦超过该阈值，复杂度函数会在新区域消失，其中临界点靠近给定向量。有趣的是，这些区域同时也包含了临界点接近多个向量的情形。