*Premise*

Let $K$ be a field of characteristic zero and $f\in K[X_1,\dots,X_m]$. By Hironaka's theorem, there exists a log resolution (over $K$) of the ideal $(f)$. Let $\{(N_i,\nu_i)\}_i$ be the numerical data of a fixed log resolution. The quantity $$ lct_K(f):=\min_{i}\frac{\nu_i}{N_i} $$ does not depend on the choice of the log resolution and it is called the log canonical threshold of $f$ over $K$.

*Questions*

Let $f\in \mathbb{Q}[X_1,\dots,X_m]$. By definition, we have $$ lct_{\mathbb{Q}}(f)\ge lct_{\mathbb{Q_p}}(f_{\mathbb{Q}_p}) \ge lct_{\mathbb{C}}(f_{\mathbb{C}}). $$ On the other hand, from Denef's formula for the motivic Igusa zeta function it follows that for all but finitely many $p$ one has $$ lct_{\mathbb{Q_p}}(f_{\mathbb{Q}_p}) \ge lct_{\mathbb{Q}}(f_{\mathbb{Q}}). $$ This shows that $$ lct_{\mathbb{Q_p}}(f_{\mathbb{Q}_p}) = lct_{\mathbb{Q}}(f_{\mathbb{Q}}) \quad \forall\forall p. $$

**1. Is this equality actually true for all $p$?**

In all the counterexamples I have found in the literature for the validity of Denef's formula for the "bad" primes (in the sense of Denef) one still has $lct_{\mathbb{Q_p}}(f_{\mathbb{Q}_p}) = lct_{\mathbb{Q}}(f_{\mathbb{Q}})$ also for bad primes $p$. Were this not always the case, has anybody a counterexample at hand?

**2. What can we say about the comparison with $lct_{\mathbb{C}}(f_{\mathbb{C}})$?**