Explain the process of polynomial curve fitting, including how error minimization is achieved, the role of the polynomial order M$,$ and how overfitting occurs. Discuss how regularization can help prevent overfitting, the impact of the dataset size on model complexity, and how techniques like least squares and maximum likelihood are used in polynomial fitting. Additionally, describe how the optimal order M or regularization parameter is selected and the relationship between model complexity and regularization. Finally, explain the effect of noise in the data and how polynomial fitting handles it$.$