On the motivic oscillation index and bound of exponential sums modulo p via analytic isomorphisms

Combining Igusa’s conjectures on exponential sums and monodromy with semicontinuity of the minimal exponent

“…These results under condition (21) imply that Conjecture 1 holds for all f in three (or less) variables. Indeed, the degree two case is easy by diagonalizing f 2 over ‫,ޑ‬ and, (21) holds when n ≤ 3 ≤ d. More surprisingly, Igusa's Conjecture (with the motivic oscillation index in the upper bound) is proved recently in [Nguyen and Veys 2022] for all polynomials in 3 variables. Some related results of the special case with n ≤ 2 are developed in [Fraser and Wright 2020;Lichtin 2013;.…”

“…The bounds for those integers N that are not squareful are recovered by the treatment of squarefree N below, by writing a general integer as a product of a squareful and a squarefree integer. We mention on the side that αf ≤ 1 if and only if f − v = 0 has nonrational singularities for some critical value v ∈ ‫ރ‬ of f , by [Saito 1993] and that in this case αf equals the minimum of the log canonical thresholds of f − v for v running over the (complex) critical values of f . These results under condition (21) imply that Conjecture 1 holds for all f in three (or less) variables.…”

“…In this section we show that Conjecture 1 with ε = 0 holds for nondegenerate polynomials, where the nondegeneracy condition is with respect to the Newton polyhedron of f − f (0) at zero as in [Castryck and Nguyen 2019] (which is slightly different than the nondegeneracy notion of [Kouchnirenko 1976;Varčenko 1976]). The nondegeneracy condition generalizes the situation where f is a sum of monomials in separate variables, like x 1 x 2 + x 3 x 4 .…”

“…, u n−1 ) = g 0 (0) = 0. Now, since f has no other possible critical value than f (0) and since there exists a toric log resolution of f − f (0) whose numerical properties (in particular its discrepancy numbers) are controlled by 0 ( f ) (see for example [Varčenko 1976]), inequality (37) follows from Igusa's work [1978]. Here, we use the following information on the discrepancy numbers coming from the toric log resolution π of f − f (0), in relation to 0 ( f ).…”

“…This hyperplane condition generalizes the case of weighted homogeneous polynomials. However, in the general case, this genericity may be lost since we imposed conditions on critical points of f τ instead of on singular points as is done more traditionally in [Kouchnirenko 1976], [Varčenko 1976]. Especially for τ = 0 ( f ) this makes a difference when the mentioned hyperplane condition is not met.…”