The classes of sums of arithmetic-geometric exponentials (SAGE) and of sums of nonnegative circuit polynomials (SONC) provide nonnegativity certificates which are based on the inequality of the arithmetic and geometric means. We study the cones ...

The Parameter Continuation Theorem is the theoretical foundation for polynomial homotopy continuation, which is one of the main tools in computational algebraic geometry. In this note, we give a short proof using Gröbner bases. Our approach gives ...

Evaluating or finding the roots of a polynomial f ( z ) = f 0 + ⋯ + f d z d with floating-point number coefficients is a ubiquitous problem. By using a piecewise approximation of f obtained with a careful use of the Newton polygon of f, we ...

In this paper we present an algorithm for efficiently counting fixed points in a finite monoid M under a conjugacy-like action. We then prove a formula for the character table of M in terms of fixed points, which allows for the effective ...

The GVW algorithm, one of the most important so-called signature-based algorithms, is designed to eliminate a large number of useless polynomial reductions from Buchberger's algorithm. The cover theorem serves as the theoretical foundation of the ...

The problem of computing the global minimum of a trigonometric polynomial is computationally hard. We address this problem for the case, where the polynomial is invariant under the exponential action of a finite group. The strategy is to follow ...

To compute the greatest common divisor (GCD) of a set of multivariate polynomials, modular algorithms are typically employed to prevent any growth in the coefficient polynomials in the intermediate expressions. However, when dealing with ...

Given two polynomials a and b in F q [ x , y ] which have no non-trivial common divisors, we prove that a generator of the elimination ideal 〈 a , b 〉 ∩ F q [ x ] can be computed in quasi-linear time. To achieve this, we propose a randomized ...

We study the boundary of the cone of real polynomials that can be decomposed as a sum of squares (SOS) of real polynomials. This cone is included in the cone of nonnegative polynomials and both cones share a part of their boundary, which ...

In this paper we construct the optimal sets of dissimilar subalgebras up to dimension three for the Lie algebra of point symmetries of the system of three-dimensional stationary equations of perfect plasticity with the Huber–von Mises yield ...

We provide an algorithm for computing a basis of homology of elliptic surfaces over P C 1 that is sufficiently explicit for integration of periods to be carried out. This allows the heuristic recovery of several algebraic invariants of the ...

We construct fundamental systems of solutions to linear ordinary differential equations, linear difference equations, and systems of partial differential equations whose elements remain linearly independent for all values of algebraically ...

The method of computing the parametric representation of an even orthogonal symplectic matrix is considered. The dimension of the family of such matrices is calculated. The general structure of matrices of small even dimensions up to 8 is ...

In his work on the twenty vertex model, Di Francesco (2021) found a determinant formula for the number of configurations in a specific such model, and he conjectured a closed form product formula for the evaluation of this determinant. We prove ...

We apply the methods of partition analysis to partitions in which the parity of parts plays a role. We begin with an in-depth treatment of the generating function for the partitions from the first Göllnitz-Gordon identity. We then deduce a ...

For a linear difference equation with the coefficients being computable sequences, we establish algorithmic undecidability of the problem of determining the dimension of the solution space including the case when some additional prior information ...

In recent years, the log-concavity of the n-th root of a sequence { S n n } n ≥ 1 has been received a lot of attention. Recently, Sun posed the following conjecture in his new book: the sequences { a n n } n ≥ 2 and { b n n } n ≥ 1 are log-...

For any zero-dimensional polynomial ideal I and any nonzero polynomial F, this paper shows that the union of the multi-set of zeros of the ideal sum I + 〈 F 〉 and that of the ideal quotient I : 〈 F 〉 is equal to the multi-set of zeros of I, where ...

In algebraic geometry, it is important to provide effective parametrizations for families of curves, both in theory and in practice. In this paper, we present such an effective parametrization for the moduli of genus-5 curves that are neither ...