Related papers: Implementation of Pellet's theorem
We prove that a bivariate polynomial f with exactly t non-zero terms, restricted to a real line {y=ax+b}, either has at most 6t-4 zeroes or vanishes over the whole line. As a consequence, we derive an alternative algorithm to decide whether…
This paper discusses the split feasibility problem with polynomials. The sets are semi-algebraic, defined by polynomial inequalities. They can be either convex or nonconvex, either feasible or infeasible. We give semidefinite relaxations…
The roots of any polynomial of degree m with complex integer coefficients can be computed by manipulation of sequences made from distinct symbols and counting the different symbols in the sequences. This method requires only primitive…
We investigate completed interlacing of zeros for pairs of polynomial sequences that fail to interlace by exactly two points. Using a general mixed recurrence relation, we identify a quadratic polynomial whose zeros serve as the two extra…
In this article, we establish necessary and sufficient conditions for a polynomial of degree $n$ to have exactly $n$ real roots. A complete study of polynomials of degree five is carried out. The results are compared with those obtained…
A polynomial is real-rooted if all of its roots are real. This note gives a simple proof of the Hermite-Sylvester theorem that a polynomial $f(x) \in {\mathbf R}[x]$ is real-rooted if and only if an associated quadratic form is positive…
In the classical theory, a famous by-product of the continued fraction expansion of quadratic irrational numbers $\sqrt{D}$ is the solution to Pell's equation for $D$. It is well-known that, once an integer solution to Pell's equation…
In this paper, pursuing the same line of ideas in the proof of an old longstanding open conjecture of \emph{Kadison-Singer} , we introduce a key lemma which we call it the interlacing lemma which indicates a necessary condition for having a…
The well-known mathematical instrument for detection common roots for pairs of polynomials and multiple roots of polynomials are resultants and discriminants. For a pair of polynomials $f$ and $g$ their resultant $R(f,g)$ is a function of…
Many proofs of the fundamental theorem of algebra rely on the fact that the minimum of the modulus of a complex polynomial over the complex plane is attained at some complex number. The proof then follows by arguing the minimum value is…
Interpolation theory for complex polynomials is well understood. In the non-commutative quaternionic setting, the polynomials can be evaluated "on the left" and "on the right". If the interpolation problem involves interpolation conditions…
The problem of writing real zero polynomials as determinants of linear matrix polynomials has recently attracted a lot of attention. Helton and Vinnikov have proved that any real zero polynomial in two variables has a determinantal…
A fundamental problem in the theory of linearized and projective polynomials over finite fields is to characterize the number of roots in the coefficient field directly from the coefficients. We prove results of this type, of a recursive…
We study in detail the zero set of a regular function of a quaternionic or octonionic variable. By means of a division lemma for convergent power series, we find the exact relation existing between the zeros of two octonionic regular…
Consider a semi-algebraic set A in R^d constructed from the sets which are determined by inequalities p_i(x)>0, p_i(x)\ge 0, or p_i(x)=0 for a given list of polynomials p_1,...,p_m. We prove several statements that fit into the following…
In this paper, we establish an analogue of the Fundamental Theorem of Algebra for polynomial matrix equations, where both the coefficient matrices and the unknown matrix are $Q$-circulant matrices. This result generalizes Abramov's result…
We introduce the concept of piecewise interlacing zeros for studying the relation of root distribution of two polynomials. The concept is pregnant with an idea of confirming the real-rootedness of polynomials in a sequence. Roughly…
Univariate polynomial root-finding is a classical subject, still important for modern computing. Frequently one seeks just the real roots of a polynomial with real coefficients. They can be approximated at a low computational cost if the…
We present a necessary and sufficient condition for a root greater than unity of a monic reciprocal polynomial of an even degree at least four, with integer coefficients, to be a Salem number. We determine the probability of fulfillment the…
We study Poncelet's Theorem in finite projective coordinate planes over the field $GF(p)$ and concentrate on a particular pencil of conics. For pairs of such conics we investigate whether we can find polygons with $n$ sides, which are…