Deterministic Shift Equivalence Testing for Sparse Polynomials

We study the shift equivalence testing problem for high-degree sparse multivariate polynomials. Given explicit spar se representations of two polynomials $f,g\in \mathbb{Q}[x_1,\dots ,x_n]$, 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 $\tau$, 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 $\mathrm{poly}(n,s,t,\tau ,\log d)$ 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 $d$.