computer algebra developers

Abstract: We study the shift equivalence testing problem for high-degree sparse multivariate polynomials. Given explicit spar se representations of two polynomials $f,g\in\mathbb{Q}[x_1,\ldots,x_n]$, with exponents encoded in binary, sparsities $s$ and $t$, and total degree at most $d$, the task is to decide whether there is a vector $\mathbf{a}\in\mathbb{Q}^n$ such that $f(\mathbf{x}+\mathbf{a})\equiv g(\mathbf{x})$. Furthermore, one needs to output such a vector if it exists. Dvir, Oliveira, and Shpilka (ICALP 2014) studied shift equivalence for low-degree arithmetic circuits. Their framew ork does not apply directly to sparse polynomials, which are not closed under shifts, and they left open whether shift equivalence of sparse polynomials admits an efficient deterministic algorithm. We answer this question positively over $\mathbb{Q}$. If the input coefficients have bit-size at most $τ$, we sh ow that a rational shift of polynomial bit-size exists whenever the polynomials are shift equivalent. Moreover, our alg orithm outputs such a shift and has a running time of $\operatorname{poly}(n,s,t,τ,\log d)$ in the bit model. The ar ithmetic and structural parts of our result also hold over any field of characteristic zero or characteristic larger th an $d$.

Abstract: We evaluate a SymPy implementation of the parallel Risch-Norman method using Charlwood's 2008 suite of 50 challenging indefinite integrals. The system automatically builds integrand towers, verifies answers by differentiation, and returns correct, verified integrals for 49 problems with zero errors. The single failure, $\int\arcsin(x\sqrt{1-x^2})\,dx$, is proven non-elementary using a holomorphic-remainder certificate, though limited by an unverified completeness hypothesis on a genus-three curve. Part II's degree bounds successfully reduce classical ansatz sizes by two-thirds without impacting running time. The paper details algorithmic mechanisms like $S'$-units, Pell units, and residue computing in tower coordinates. Compared to mature implementations, this untuned SymPy prototype is slower than FriCAS (by a factor of six on the median integral) but more accurate than AXIOM (which returned three wrong answers). Profiling pinpoints performance bottlenecks in nonlinear norm searches and nested number fields, outlining clear targets for optimisation.