Computational Approaches to the Singer Transfer: Preimages in the Lambda Algebra and $G_k$-Invariant Theory

We present a systematic, algorithmic method to compute the preimage of elements under the Singer algebraic transfer. Using the lambda algebra and the invariant-theoretic formula of P.H. Chon and L.M. Ha [5], we formulate the preimage search as a solvable problem in linear algebra. This framework is applied to study key indecomposable elements in the Adams spectral sequence. As a consequence, we show that the proof of the known result that the indecomposable element $d_0\in {\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathcal{A}}^{4,18}(\mathbb{Z}\mathrm{/}2,\mathbb{Z}\mathrm{/}2)\mathrm{.}$ lies in the image of the fourth Singer transfer, as given by Nguyen Sum in [17], is false. Furthermore, we provide the explicit description of a preimage for the indecomposable element $p_0\in {\mathrm{E}\mathrm{x}\mathrm{t}}_{\mathcal{A}}^{4,37}(\mathbb{Z}\mathrm{/}2,\mathbb{Z}\mathrm{/}2)\mathrm{.}$ This preimage had not been explicitly determined in the previous work of N.H.V. Hung and V.T.N. Quynh [8]. Finally, our most significant contribution is the construction of a complete $\text{textsc}SageMath$ algorithm that fully automates the computation of both the dimension and an explicit basis for the $G_k$-invariant space $[(Q{\mathcal{P}}_k{)}_d{]}^{G_k}$. This tool facilitates the verification of our results [11, 12, 13] that were previously computed manually in connection with Singer’s conjecture for rank 4.