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A problem concerning the shift of roots of a system of homogeneous algebraic equations is investigated. Its conservation and decomposition of a multiple root into simple roots are discussed.

This note presents the multivariate Hermite criterion: a practical and powerful algorithm for determining the number of distinct real and complex roots of a zero-dimensional system of polynomials in any finite number of variables. The final…

An exact, number-conserving solution to the generalized, orbit-dependent pairing problem is derived by introducing an infinite-dimensional algebra. A method for obtaining eigenvalues and eigenvectors of the corresponding Hamiltonian is also…

In this paper, we extend the work of (Abbondati et al., 2024) on decoding simultaneous rational number codes by addressing two important scenarios: multiplicities and the presence of bad primes (divisors of denominators). First, we…

We follow a stream of the history of positive matrices and positive functionals, as applied to algebraic sums of squares decompositions, with emphasis on the interaction between classical moment problems, function theory of one or several…

We present an exact and complete algorithm to isolate the real solutions of a zero-dimensional bivariate polynomial system. The proposed algorithm constitutes an elimination method which improves upon existing approaches in a number of…

Many problems give rise to polynomial systems. These systems often have several parameters and we are interested to study how the solutions vary when we change the values for the parameters. Using predictor-corrector methods we track the…

We apply verified numerics to the Nirenberg problem, proving that a genuine solution exists near two given computer-generated approximate solutions. This proves existence of a solution for a particular prescribed curvature that was…

We present a general method for constructing real solutions to some problems in enumerative geometry which gives lower bounds on the maximum number of real solutions. We apply this method to show that two new classes of enumerative…

Complex polynomial optimization has recently gained more and more attention in both theory and practice. In this paper, we study the optimization of a real-valued general conjugate complex form over various popular constraint sets including…

Sturm's theorem (1829/35) provides an elegant algorithm to count and locate the real roots of any real polynomial. In his residue calculus (1831/37) Cauchy extended Sturm's method to count and locate the complex roots of any complex…

We present some new and recent algorithmic results concerning polynomial system solving over various rings. In particular, we present some of the best recent bounds on: (a) the complexity of calculating the complex dimension of an algebraic…

Bialgebras and Hopf (bi)modules are typical algebraic structures with several interacting operations. Their structural and homological study is therefore quite involved. We develop the machinery of braided systems, tailored for handling…

This article is divided in two parts. In the first part we review some recent results concerning the expected number of real roots of random system of polynomial equations. In the second part we deal with a different problem, namely, the…

The radical solution of polynomials with rational coefficients is a famous solved problem. This paper found that it is a $\mathbb{NP}$ problem. Furthermore, this paper found that arbitrary $ \mathscr{P} \in \mathbb{P}$ shall have a one-way…

Those lectures revolve around the following problem: given a system of n real polynomials in n variables, count the number of real roots. The first lecture is a course on Newton iteration and alpha-theory. The second describes an…

This paper solves the open problem on the sharp bound for the number of isolated solutions in $\mathbf{R}_*^n$ to the real system of $n$ polynomial equations in $n$ variables, i.e., the real $n$ by $n$ fewnomial system. For an unmixed…

The Waring Problem over polynomial rings asks for how to decompose an homogeneous polynomial of degree $d$ as a finite sum of $d^{th}$ powers of linear forms. First, we give a constructive method to obtain a real Waring decomposition of any…

A classification of ordinary differential equations and finite-difference equations in one variable having polynomial solutions (the generalized Bochner problem) is given. The method used is based on the spectral problem for a polynomial…

In this work we study the space complexity of computable real numbers represented by fast convergent Cauchy sequences. We show the existence of families of trascendental numbers which are logspace computable, as opposed to algebraic…