基于E_8格的θ级数的第8次根和Leech格的θ级数的第24次根都具有整数系数的发现，我们研究了R中任意元素f（其中R = 1 + xZ[[x]]）何时可以写成f = g^n，其中g在R中，n >= 2.设P_n：= {g^n：g in R}，设mu_n：= n Product_{p|n} p.我们证明了（i）对f in R，f in P_n <=> fmod mu_n in P_n，（ii）若f in P_n，则P_n中存在系数为mod mu_n/n的唯一g，使得f == g^n（mod mu_n）.特别地，如果f == 1（mod mu_n），则f在P_n中.后一个断言意味着R^n中的任何极值偶幺模格（例如R^8中的E_8）的θ级数在P_n中，如果n的形式为2^i3^j5^k（i ≥ 3）。虽然我们证明了长度为2^m的r阶Reed-Muller码的重量计数器在P_{2^r}中，但似乎没有任何精确的码的类似物。我们给出了一些其它的结果和定理，并建立了Paul D. Hanna证明了P_n（n！= 1）中存在唯一的元素f 2)其中系数被限制为集合{1，2，...，n}。
英文摘要：
Motivated by the discovery that the eighth root of the theta series of the E_8 lattice and the 24th root of the theta series of the Leech lattice both have integer coefficients, we investigate the question of when an arbitrary element f in R (where R = 1 + xZ[[x]]) can be written as f = g^n for g in R, n >= 2. Let P_n := {g^n : g in R} and let mu_n := n Product_{p|n} p. We show among other things that (i) for f in R, f in P_n <=> f mod mu_n in P_n, and (ii) if f in P_n, there is a unique g in P_n with coefficients mod mu_n/n such that f == g^n (mod mu_n). In particular, if f == 1 (mod mu_n) then f in P_n. The latter assertion implies that the theta series of any extremal even unimodular lattice in R^n (e.g. E_8 in R^8) is in P_n if n is of the form 2^i 3^j 5^k (i >= 3). There do not seem to be any exact analogues for codes, although we show that the weight enumerator of the r-th order Reed-Muller code of length 2^m is in P_{2^r}. We give a number of other results and conjectures, and establish a conjecture of Paul D. Hanna that there is a unique element f in P_n (n != 2) with coefficients restricted to the set {1, 2, ..., n}.
Intégralité des coefficients de Taylor de racines d'applications miroir
证明了镜像映射根的泰勒系数在原点处的可积性。镜像映射是指形式幂级数z.exp（G（z）/F（z）），其中F（z）和G（z）+log（z）F（z）是某些广义超几何微分方程的特解.这使我们能够证明一个猜想所述的周“积分性质的变化的马勒措施”[arXiv：1006.2428v1数学。AG]。这些结果的证明是在我们的文章中使用的技术的适应“关键pour l 'intégralité des coefficients de Taylor des applications miroir”，[J. Reine Angew.数学（出现）。
The generating function for restricted partitions is a finite product with a Laurent expansion at each root of unity. The question of the behavior of these Laurent coefficients as the size of the product increases goes back to Rademacher and his work on partitions. Building on the methods of Drmota, Gerhold and previous results of the author, we complete this description and give the full asymptotic expansion of each coefficient at every root of unity. These techniques are also shown to give the asymptotics of Sylvester waves.
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In 2007, G.E. Andrews introduced the $(n+1)$-variable combinatorial generating function $R_n(x_1,x_2,\cdots,x_n;q)$ for ranks of $n$-marked Durfee symbols, an $(n+1)$-dimensional multisum, as a vast generalization to the ordinary two-variable partition rank generating function. Since then, it has been a problem of interest to understand the automorphic properties of this function; in special cases and under suitable specializations of parameters, $R_n$ has been shown to possess modular, quasimodular, and mock modular properties when viewed as a function on the upper half complex plane $\mathbb…
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Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact for a univariate polynomial $f$. Our proof is constructive as it gives a closed-form expression for the number of roots $N_{k}(f)$. Our proof, when combined with the recent root-counting algorithm of (Dwivedi, Mittal, Saxena, CCC, 2019), yields the first deterministic poly($|f|, \log …
