关于产生极大非结合拟群的二次正交态射的数量
On the number of quadratic orthomorphisms that produce maximally nonassociative quasigroups
AI总结：
该研究针对奇素数幂q下由二次运算定义的拟群Q_{a,b}，推导了使其为极大非结合拟群的参数对(a,b)的数量σ(q)的渐近极限，给出q模4不同余时的两个常数α、β。
摘要：
设q为奇素数幂，且a,b属于有限域F_q，满足ab和(1-a)(1-b)均为非零平方元。令Q_{a,b} = (F_q, *)为拟群，其运算定义为：若v-u是平方元，则u*v = u + a(v-u)；若v-u是非平方元，则u*v = u + b(v-u)。若该拟群满足x*(y*z) = (x*y)*z当且仅当x=y=z，则称其为极大非结合拟群。记σ(q)为使得Q_{a,b}为极大非结合拟群的(a,b)的数量。我们证明存在常数α≈0.02908和β≈0.01259，使得当q≡1 mod 4时，极限σ(q)/q² = α；当q≡3 mod 4时，极限σ(q)/q² = β。
英文摘要：
Let $q$ be an odd prime power and suppose that $a,b\in\mathbb{F}_q$ are such that $ab$ and $(1{-}a)(1{-}b)$ are nonzero squares. Let $Q_{a,b} = (\mathbb{F}_q,*)$ be the quasigroup in which the operation is defined by $u*v=u+a(v{-}u)$ if $v-u$ is a square, and $u*v=u+b(v{-}u)$ is $v-u$ is a nonsquare. This quasigroup is called maximally nonassociative if it satisfies $x*(y*z) = (x*y)*z$ $\Leftrightarrow$ $x=y=z$. Denote by $σ(q)$ the number of $(a,b)$ for which $Q_{a,b}$ is maximally nonassociative. We show that there exist constants $α\approx 0.02908$ and $β\approx 0.01259$ such that if $q\equiv 1 \bmod 4$, then $\lim σ(q)/q^2 = α$, and if $q \equiv 3 \bmod 4$, then $\lim σ(q)/q^2 = β$.
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