[数学] 特征二中的一族新的例外多项式 [推广有奖]

摘要翻译：
在特征为2的域K上，我们产生了一个新的多项式族f(x)，它是例外的，即除了x-y的标量倍数外，在K[x，y]中f(x)-f(y)没有绝对不可约因子；当K是有限的时，这个条件等价于说有无穷多个有限扩张L/K，其映射C-->f(c)在L上是双射的。我们的多项式有次(2^e-1)*2^(e-1)，其中e是奇数。结合文献ARXIV:0707.1835，完成了不可分解的非特征幂次例外多项式的分类。我们的证明策略是识别由例外多项式F诱导的分支覆盖P^1-->P^1的Galois闭包的曲线。在这种情况下，曲线是x^(q+1)+y^(q+1)=a+T(xy)，其中T(z)=z^(q/2)+z^(q/4)+...+z。我们的证明依赖于曲线的Galois复盖中分支的新性质，以及某些2-参数族中所有曲线的自同构群的计算。
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英文标题：
《A new family of exceptional polynomials in characteristic two》
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作者：
Robert M. Guralnick, Joel E. Rosenberg and Michael E. Zieve
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最新提交年份：
2008
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分类信息：

一级分类：Mathematics        数学
二级分类：Number Theory        数论
分类描述：Prime numbers, diophantine equations, analytic number theory, algebraic number theory, arithmetic geometry, Galois theory
素数，丢番图方程，解析数论，代数数论，算术几何，伽罗瓦理论
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一级分类：Mathematics        数学
二级分类：Algebraic Geometry        代数几何
分类描述：Algebraic varieties, stacks, sheaves, schemes, moduli spaces, complex geometry, quantum cohomology
代数簇，叠，束，格式，模空间，复几何，量子上同调
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英文摘要：
  We produce a new family of polynomials f(x) over fields K of characteristic 2 which are exceptional, in the sense that f(x)-f(y) has no absolutely irreducible factors in K[x,y] besides the scalar multiples of x-y; when K is finite, this condition is equivalent to saying there are infinitely many finite extensions L/K for which the map c --> f(c) is bijective on L. Our polynomials have degree (2^e-1)*2^(e-1), where e is odd. Combined with our previous paper arxiv:0707.1835, this completes the classification of indecomposable exceptional polynomials of degree not a power of the characteristic. The strategy of our proof is to identify the curves that can arise as the Galois closure of the branched cover P^1 --> P^1 induced by an exceptional polynomial f. In this case, the curves turn out to be x^(q+1)+y^(q+1)=a+T(xy), where T(z)=z^(q/2)+z^(q/4)+...+z. Our proofs rely on new properties of ramification in Galois covers of curves, as well as the computation of the automorphism groups of all curves in a certain 2-parameter family.
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PDF链接：
https://arxiv.org/pdf/0707.1837