Is polynomial regression typically linear or non-linear?

Question

Here's my doubt:

When we say polynomial regression, do we usually mean linear or non-linear?

A simple answer would be very helpful.

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_{^{This is not a duplicate question. I've read through similar questions and require clarification on the nuances of the terminology.}}

Answer 1

Polynomial regression usually stands for the fitting of a polynomial to a set of $(x,y)$ points, which is a linear problem when solved by least-squares.

Answer 2

No. As was already pointed out - the linearity in regression model is defined w.r.t. the parameters $\beta$ and not the variables $x$. Formally, if the gradient vector of (the conditional expectation of the model) is independent of any unknown terms - then it is linear, otherwise non-linear. Namely, for a polynomial regression $$ y_i = \beta_0 + \sum_j^p \beta_j x_{i}^ j + \epsilon_i, $$ the gradient w.r.t. the $\beta$ is $$ \nabla_{\beta} \mathbb{E}[y|x]=(1, x, x^2,...,x^p)^T, $$ that is independent of any unknown factors, hence the model is "linear".