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Many quantum computing workflows manipulate long lists of Pauli strings. A basic classical subroutine involves taking $m$ Pauli strings on $n$ qubits, each of weight bounded by a constant, to determine if they are pairwise commuting,…
One of the major promises of quantum computing is the realization of SIMD (single instruction - multiple data) operations using the phenomenon of superposition. Since the dimension of the state space grows exponentially with the number of…
We conjecture that bounded generalised polynomial functions cannot be generated by finite automata, except for the trivial case when they are ultimately periodic. Using methods from ergodic theory, we are able to partially resolve this…
The resultant of two univariate polynomials is an invariant of great importance in commutative algebra and vastly used in computer algebra systems. Here we present an algorithm to compute it over Artinian principal rings with a modified…
A semiprime is a natural number which can be written as the product of two primes. The asymptotic behaviour of the function $\pi_2(x)$, the number of semiprimes less than or equal to $x$, is studied. Using a combinatorial argument,…
We study polynomial identities of nonassociative algebras constructed by using infinite binary words and their combinatorial properties. Infinite periodic and Sturmian words were first applied for constructing examples of algebras with…
A classical problem in Distance Geometry, with multiple practical applications (in molecular structure determination, sensor network localization etc.) is to find the possible placements of the vertices of a graph with given edge lengths.…
We develop algorithms for writing a polynomial as sums of powers of low degree polynomials. Consider an $n$-variate degree-$d$ polynomial $f$ which can be written as $$f = c_1Q_1^{m} + \ldots + c_s Q_s^{m},$$ where each $c_i\in…
When the Euclidean algorithm produces a symmetric sequence of quotients, we give explicit formulas for the remainders that allow the analysis of two families of quadratic forms in the remainders.
We present a quasilinear time algorithm to decide the word problem on a natural algebraic structures we call orthocomplemented bisemilattices, a subtheory of boolean algebra. We use as a base a variation of Hopcroft, Ullman and Aho…
In the monotone integer dualization problem, we are given two sets of vectors in an integer box such that no vector in the first set is dominated by a vector in the second. The question is to check if the two sets of vectors cover the…
In many data analysis pipelines, a basic and time-consuming process is to produce join results and feed them into downstream tasks. Numerous enumeration algorithms have been developed for this purpose. To be a statistically meaningful…
Formulas involving fundamental mathematical constants had a great impact on various fields of science and mathematics, for example aiding in proofs of irrationality of constants. However, the discovery of such formulas has historically…
We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In…
The original F5 algorithm introduced by Faug\`ere is formulated for any homogeneous polynomial set input. The correctness of output is shown for any input that terminates the algorithm, but the termination itself is proved only for the case…
Nested sums containing binomial coefficients occur in the computation of massive operator matrix elements. Their associated iterated integrals lead to alphabets including radicals, for which we determined a suitable basis. We discuss…
We develop a new algorithm for factoring a bivariate polynomial $F\in \mathbb{K}[x,y]$ which takes fully advantage of the geometry of the Newton polygon of $F$. Under a non degeneracy hypothesis, the complexity is…
Recently several authors have proved results on Ehrhart series of free sums of rational polytopes. In this note we treat these results from an algebraic viewpoint. Instead of attacking combinatorial statements directly, we derive them from…
This paper deals with lattice congruences of the weak order on the symmetric group, and initiates the investigation of the cover graphs of the corresponding lattice quotients. These graphs also arise as the skeleta of the so-called…
We consider systems of recursively defined combinatorial structures. We give algorithms checking that these systems are well founded, computing generating series and providing numerical values. Our framework is an articulation of the…