AI 中文总结
研究在素域\(\mathbb{F}_p\)（\(p>2\)）上，通过用系数矩阵的\(2\times2\)子式表示朗斯基行列式输出，并从一阶子结式和线性合冲关系恢复，给出无导数的里歇洛\((2,2)\)同源重述，还介绍了相关认证方法及速度提升。
AI 中文摘要
亏格为\(2\)的曲线的经典里歇洛\((2,2)\)同源步骤通过朗斯基导数从因式分解\(f = uvw\)构建一个陪域三元组\((U,V,W)\)。我们在素域\(\mathbb{F}_p\)（\(p>2\)）上给出了一个完全无导数的重新表述，通过系数矩阵的\(2\times2\)子式来表示朗斯基行列式输出，并从一阶子结式和一个线性合冲关系中恢复它们。由此产生的余数多项式路径（RPR）被证明在\(\mathbb{F}_p[x]\)中产生相同的输出三元组，不仅是在单位元意义下，而且是作为一个精确的多项式恒等式。基于这种等价性，我们引入了受保护的子结式路径（GSR），一种确定性评估器，它通过常数大小的代数保护、轻量级后检查以及最多一次有界仿射重试来认证可接受性。所有路径每步执行\(O(1)\)次域运算。在每个素数上进行\(10^6\)次匹配试验的原型证实，RPR比经典的朗斯基行列式公式的内核速度快\(4.75 - 6\)倍，并且尽管有认证开销，完整的GSR管道仍然比WRO快\(1.4 - 3\)倍。通过对五个素数上的\(2.5\times10^5\)个随机三元组进行双里歇洛对合测试，独立验证了正确性。
英文摘要
The Richelot $(2,2)$-step is the standard step of explicit genus-2 isogeny computation. We determine the exact stratification of its input space over a finite field $\mathbb{F}_q$ of odd characteristic: ordered factorizations $f=uvw$ of a square-free sextic into monic quadratics fall into three strata by the geometric type of the quotient, governed by the incidence geometry of the discriminant locus. This yields closed formulas for each stratum and, modulo affine coordinate changes, mass formulas of degree four in $q$ with a complete classification of stabilisers. The classification is decided by data the step already computes, at $5\mathrm{M}+6\mathrm{S}$ beyond the brackets, and an output post-check is provably redundant. On the decomposable stratum the square class of one resultant determines the field of definition of the elliptic factors and the shape of the Weil polynomial of the Jacobian; the two cases are counted exactly, and in the nonsplit case the curve $y^2=f$ has $q+1$ rational points. Exhaustive enumeration over small finite fields verifies every proved count.
CommentsSubstantially revised and extended; supersedes v1, then titled "Derivative-free Richelot isogenies via subresultants with algebraic certification". Corrects v1 and adds exact stratification counts, affine mass formulas with a full stabiliser classification, and the base-field arithmetic of the decomposable stratum. 46 pages; verification scripts included as ancillary files
