Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On a uniform bound for exponential sums modulo $p^m$ for Deligne
  polynomials   Let $f$ be a polynomial of degree $d>1$ in $n$ variables over $\mathbb{Z}$.
Let $f_d$ be the homogeneous part of degree $d$ of $f$ and $s$ be the dimension
of the critical locus of $f_d$. In this paper, we prove Igusa's conjecture for
exponential sums with the exponent $(n-s)/(2(d-1))$. This implies a weak
solution for a recent conjecture raised by Cluckers and the author (2020) about
an analogue of the results of Deligne (1974) and Katz (1999) for exponential
sums over finite fields in the finite ring setting. Moreover, this also
improves the result of Cluckers, Musta\c{t}\u{a} and the author (2019) in case
$n-s>2(d-1)$. In particular, this result improves the conditions $n-s>2^d(d-1)$
of Birch (1962) and $n-s>3(d-1)2^{d-2}$ of Browning-Prendiville (2017) on the
validity of the estimation for the major arcs to $(n-s)>4(d-1)$. Therefore this
result may have further applications on subjects related to the
Hardy-Littlewood circle method such as the Hasse principle or distribution of
rational points in algebraic varieties. On the other hand, we also improve the
recent work of Cluckers, Koll\'ar and Musta\c{t}\u{a} (2019) on the strong
monodromy conjecture in the range $(-{\rm lct}((f)+J_f^2),0]$ in case of bad
reduction and bad Schwartz-Bruhat function. Namely, in the range $(-{\rm
lct}((f)+J_f^2),0]$, the real part of any pole of the Igusa local zeta
functions associated with $f$ and any Schwartz-Bruhat function over any
$p$-adic field is a root of the Bernstein-Sato polynomial of $f$.

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The categories of this paper are: acc-phys
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
4
The categories of this paper are: cond-mat
15
The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
34
The categories of this paper are: chem-ph
7
The categories of this paper are: hep-th
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The categories of this paper are: mtrl-th
33
The categories of this paper are: plasm-ph
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The categories of this paper are: dg-ga
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The categories of this paper are: patt-sol
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The categories of this paper are: cs
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The categories of this paper are: alg-geom
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The categories of this paper are: hep-lat