Blow-up Algorithm for Sum-of-Products Polynomials and Real Log Canonical Thresholds

When considering an invariant that gives a Bayesian generalization error, that is a real log canonical threshold, in general, papers replace a mean error function with a relatively simple polynomial whose real log canonical threshold corresponds to that of the mean error function, and obtain its real log canonical threshold by resolving its singularities through an algebraic operation called blow-up. Though it is known that the singularities of any polynomial can be resolved by a finite number of blow-up iterations, it is not clarified well whether or not it is possible to resolve singularities of a specific polynomial by applying a specific blow-up algorithm. Therefore this paper proposes a blow-up algorithm that can be applied to the polynomials called sum-of-products polynomials and proves that it halts. Furthermore, this paper considers real log canonical thresholds of sum-of-products polynomials by using the algorithm. First, this section explains the foundation of Bayesian learning theory and details the relation to a real log canonical threshold and blow-up. Then this section defines exclusive sum-of-products polynomials which is subject to previous studies and explains the novelty and utility of this paper.