Cross-Characteristic Gate Complexity of the Algebraic Torus
Yipin Wang
ECCC, 2026
We determine the gate complexity $t(p,q,n)$ for all primes $p$ and prime powers $q$ with $\mathrm{char}(\mathbb{F}_q) \neq p$.
A dichotomy emerges: $t(p,q,n) = (q-1)^{n-1}$ when $p \mid (q-1)$, and
$t(p,q,n) = (q^n - 1)/(q-1)$ when $p \nmid (q-1)$.
The lower bound uses a Frobenius orbit counting argument over $\mathbb{F}_{p^k}$,
exploiting the self-duality $\widehat{\mathbf{1}_T} = \mathbf{1}_T$.
We give a cohomological interpretation: gate complexity equals the Frobenius trace
on $H^*_c(\mathbb{G}_m^{n-1}, \mathbb{F}_p)$.
A Fourier-Analytic Switching Lemma over $\mathbb{F}_p$ and the AC$^0$ Lower Bound for Generalized Parity
Yipin Wang
ECCC, 2026
We prove a switching lemma for constant-depth circuits over $\mathbb{F}_p$
with generalized AND/OR gates, extending Tal's Fourier-analytic approach from the Boolean setting.
The key ingredient is an exact computation of the $L_1$ Fourier mass of AND/OR gates over $\mathbb{F}_p$.
As a consequence, constant-depth circuits of sub-exponential size
over $\mathbb{F}_p$ cannot compute $\mathbf{1}[\sum_i x_i \equiv 0 \pmod{p}]$.
A Switching Lemma for DNFs over $\mathbb{F}_p$: the Canonical Decision Tree Approach
Yipin Wang
Preprint, 2026
We prove an $M$-independent switching lemma for width-$w$ DNFs over $\mathbb{F}_p$ via the canonical decision tree method:
$\Pr[\mathrm{DT}(f|_\rho) \geq s] \leq (2pwq/(1-q))^s$, with no dependence on the number of terms.
The key observation is that the $p$-ary branching of the CDT is effectively binary,
since all $p-1$ nonzero values have the same effect on literal satisfaction.
Gibbs Intertwining Operators and the Steinberg Polynomial
Yipin Wang
Preprint, 2026
We study the Markov operator $P$ on $\mathbb{P}^1(\mathbb{F}_p)$ with Gibbs weights $w_r = q^{p-r}/(q^p - 1)$,
showing it factors as a deformed intertwining operator $L_w \cdot \pi(w_0)$ that does not belong to the
Iwahori–Hecke algebra. The main result is a twisted circulant reduction:
$n_p(q) = -(q-1)(q^p-1)\det(I - C)$, where $C = Q \cdot W$ on $\mathbb{F}_p^*$ mixes the additive
convolution $W$ with multiplicative inversion $Q\colon j \mapsto j^{-1}$. We establish a polarization
identity $W^* + W = I$ on the unit circle and prove it descends to the block diagonal entries of $C = QW$.
We prove an exact Cayley determinantal identity $\det(I-C) = \det(I+U-Q)/\det(I+U)$ and discover that
$R = QUQU^{-1}$ is a $q$-independent permutation of order 3, giving an $S_3$ action on the spectral data.
The Steinberg Polynomial: Endoscopic Decomposition and Motivic Factorization
Yipin Wang
Preprint, 2026
We introduce the Steinberg polynomial $n_p(q) = (q^p - 1)\det(I - P|_{\mathrm{St}_p}) \in \mathbb{Z}[q]$
and show it admits an endoscopic decomposition
$n_p = n_p^{\mathrm{GL}_2} - \bigl(\frac{-2}{p}\bigr) n_p^T$
into palindromic and anti-palindromic parts, together with a motivic factorization into CM abelian
varieties over $\mathbb{F}_2$ with CM by subfields of $\mathbb{Q}(\sqrt{-2}, \zeta_p)$.
We define alien primes as odd primes $\ell > 3$ dividing $n_p(2)$ but not $p(2^p - 1)$,
and prove they are disjoint from Hecke discriminant primes.
Spanning Trees, Modular Symbols, and a New Arithmetic Invariant of Elliptic Curves
Yipin Wang
Preprint, 2026
Averaging the plus modular symbol $\{0, t/u\}^+$ over spanning-tree-weighted feasible vectors
gives a rational invariant $c_f \in \mathbb{Q}$ for each weight-2 newform $f$.
The proof reduces to a Markov chain on $\mathbb{P}^1(\mathbb{Z}/N\mathbb{Z})$.
We compute $c_f$ for all optimal curves of conductor $\leq 100$ and study the
prime structure of its denominators.
On the 2-adic Structure of Zagier's MZV Matrices
Yipin Wang
Preprint, 2026
We prove that all coefficients in the decomposition of $\zeta(2)^{K-1}\zeta(3)$ into the
Hoffman basis are odd integers, and establish a row minimum formula for the 2-adic
valuation of $(M_K)^{-1}$. As a companion result, we give a closed-form inverse for the
binomial core matrix $B_N[a,i] = \binom{2i}{2a}$ in terms of Euler–secant numbers.