Polynomial kernel

The problem, from first principles

Linear kernel gives straight boundaries, and RBF kernel gives bumpy ones driven by distance. A polynomial kernel sits between them: it raises a shifted, scaled dot product to a power, so the boundary can bend in controlled, polynomial ways. Degree 1 is the linear kernel again, and higher degrees allow more curvature.

Given X with shape (n, d) and Y with shape (m, d), return the (n, m) matrix of polynomial similarities.

Constraints

• Return an array of shape (n, m) where entry [i, j] equals (gamma * X[i] @ Y[j] + coef0) ** degree.
• Defaults: degree=3, coef0=1.0, gamma=1.0.
• Build it on top of linear_kernel from the previous question, which is already available to load.
• Do not modify X or Y.

Hints

Open one at a time. Each gives away a little more than the last.