We prove that for polynomials $ f, g, h \in \mathbb {Z}[x] $ satisfying $ f = gh $ and $ f(0) \neq 0 $, the $\ell _2$-norm of the cofactor $ h $ is bounded by

$$ \begin{align*} \left\Vert {h} \right\Vert{}_2\leq \sqrt{\frac{\left\Vert {g} \right\Vert{}_0}{\deg g}}\cdot \left\Vert {f} \right\Vert{}_1\cdot \left(\widetilde{O}\left( \frac{\left\Vert {g} \right\Vert{}_0^{2.5} \cdot \deg^2{f}}{\sqrt{\deg{g}}} \right) \right)^{\left\Vert {g} \right\Vert{}_0-1} \ , \end{align*} $$

where $\left \Vert {g} \right \Vert {}_0$ is the number of nonzero coefficients of g (its sparsity). We also obtain similar results for polynomials over $\mathbb {C}$. These bounds are an improvement over the bounds presented in an earlier conference version of this paper [NS24].

This result significantly improves upon previously known exponential bounds (in $\deg {f}$) for general polynomials. It further implies that, under exact division, the polynomial division algorithm runs in quasi-linear time with respect to the input size and the number of terms in the quotient $ h $. This resolves a long-standing open problem concerning the exact divisibility of sparse polynomials.

In particular, our result demonstrates a quadratic separation between the runtime (and representation size) of exact and nonexact divisibility by sparse polynomials. Notably, prior to our work, it was not even known whether the representation size of the quotient polynomial could be bounded by a sub-quadratic function of its number of terms, or even by a subquadratic function of $\deg {f}$.

We consider the relationship between the Mahler measure $M(f)$ of a polynomial f and its separation $\operatorname {sep}(f)$. Mahler [‘An inequality for the discriminant of a polynomial’, Michigan Math. J. 11 (1964), 257–262] proved that if $f(x) \in \mathbb {Z}[x]$ is separable of degree n, then $\operatorname {sep}(f) \gg _n M(f)^{-(n-1)}$. This spurred further investigations into the implicit constant involved in that relationship and led to questions about the optimal exponent on $M(f)$. However, there has been relatively little study concerning upper bounds on $\operatorname {sep}(f)$ in terms of $M(f)$. We prove that if $f(x) \in \mathbb {C}[x]$ has degree n, then $\operatorname {sep}(f) \ll n^{-1/2}M(f)^{1/(n-1)}$. Moreover, this bound is sharp up to the implied constant factor. We further investigate the constant factor under various additional assumptions on $f(x)$; for example, if it has only real roots.

We study the geometry of tropical extensions of hyperfields, including the ordinary, signed, and complex tropical hyperfields. We introduce the framework of ‘enriched valuations’ as hyperfield homomorphisms to tropical extensions and show that a notable family of them are relatively algebraically closed. Our main results are hyperfield analogues of Kapranov’s theorem and the Fundamental theorem of tropical geometry. Utilizing these theorems, we introduce fine tropical varieties and prove a structure theorem for them in terms of their initial ideals.

We provide a criterion for a coherent sheaf to be an Ulrich sheaf in terms of a certain bilinear form on its global sections. When working over the real numbers, we call it a positive Ulrich sheaf if this bilinear form is symmetric or Hermitian and positive-definite. In that case, our result provides a common theoretical framework for several results in real algebraic geometry concerning the existence of algebraic certificates for certain geometric properties. For instance, it implies Hilbert’s theorem on nonnegative ternary quartics, via the geometry of del Pezzo surfaces, and the solution of the Lax conjecture on plane hyperbolic curves due to Helton and Vinnikov.

For any real polynomial $p(x)$ of even degree k, Shapiro [‘Problems around polynomials: the good, the bad and the ugly$\ldots $’, Arnold Math. J. 1(1) (2015), 91–99] proposed the conjecture that the sum of the number of real zeros of the two polynomials $(k-1)(p{'}(x))^{2}-kp(x)p{"}(x)$ and $p(x)$ is larger than 0. We prove that the conjecture is true except in one case: when the polynomial $p(x)$ has no real zeros, the derivative polynomial $p{'}(x)$ has one real simple zero, that is, $p{'}(x)=C(x)(x-w)$, where $C(x)$ is a polynomial with $C(w)\ne 0$, and the polynomial $(k-1)(C(x))^2(x-w)^{2}-kp(x)C{'}(x)(x-w)-kC(x)p(x)$ has no real zeros.

We study multivariate polynomials over ‘structured’ grids. Firstly, we propose an interpretation as to what it means for a finite subset of a field to be structured; we do so by means of a numerical parameter, the nullity. We then extend several results – notably, the Combinatorial Nullstellensatz and the Coefficient Theorem – to polynomials over structured grids. The main point is that the structure of a grid allows the degree constraints on polynomials to be relaxed.

We solve the problem of factoring polynomials $V_n(x) \pm 1$ and $W_n(x) \pm 1$, where $V_n(x)$ and $W_n(x)$ are Chebyshev polynomials of the third and fourth kinds, in terms of the minimal polynomials of $\cos ({2\pi }{/d})$. The method of proof is based on earlier work, D. A. Wolfram, [‘Factoring variants of Chebyshev polynomials of the first and second kinds with minimal polynomials of $\cos ({2 \pi }/{d})$’, Amer. Math. Monthly 129 (2022), 172–176] for factoring variants of Chebyshev polynomials of the first and second kinds. We extend this to show that, in general, similar variants of Chebyshev polynomials of the fifth and sixth kinds, $X_n(x) \pm 1$ and $Y_n(x) \pm 1$, do not have factors that are minimal polynomials of $\cos ({2\pi }/{d})$.

Without resorting to complex numbers or any advanced topological arguments, we show that any real polynomial of degree greater than two always has a real quadratic polynomial factor, which is equivalent to the fundamental theorem of algebra. The proof uses interlacing of bivariate polynomials similar to Gauss’s first proof of the fundamental theorem of algebra using complex numbers, but in a different context of division residues of strictly real polynomials. This shows the sufficiency of basic real analysis as the minimal platform to prove the fundamental theorem of algebra.

A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].

The aim of this note is to give a simple topological proof of the well-knownresult concerning continuity of roots of polynomials. We also consider amore general case with polynomials of a higher degree approaching a givenpolynomial. We then examine the continuous dependence of solutions of lineardifferential equations with constant coefficients.

We upper-bound the number of common zeros over a finite grid of multivariate polynomials and an arbitrary finite collection of their consecutive Hasse derivatives (in a coordinate-wise sense). To that end, we make use of the tool from Gröbner basis theory known as footprint. Then we establish and prove extensions in this context of a family of well-known results in algebra and combinatorics. These include Alon's combinatorial Nullstellensatz [1], existence and uniqueness of Hermite interpolating polynomials over a grid, estimations of the parameters of evaluation codes with consecutive derivatives [20], and bounds on the number of zeros of a polynomial by DeMillo and Lipton [8], Schwartz [25], Zippel [26, 27] and Alon and Füredi [2]. As an alternative, we also extend the Schwartz-Zippel bound to weighted multiplicities and discuss its connection to our extension of the footprint bound.

Let $R\subset F$ be an extension of real closed fields, and let ${\mathcal{S}}(M,R)$ be the ring of (continuous) semialgebraic functions on a semialgebraic set $M\subset R^{n}$. We prove that every $R$-homomorphism ${\it\varphi}:{\mathcal{S}}(M,R)\rightarrow F$ is essentially the evaluation homomorphism at a certain point $p\in F^{n}$ adjacent to the extended semialgebraic set $M_{F}$. This type of result is commonly known in real algebra as a substitution lemma. In the case when $M$ is locally closed, the results are neat, while the non-locally closed case requires a more subtle approach and some constructions (weak continuous extension theorem, appropriate immersion of semialgebraic sets) that have interest of their own. We consider the same problem for the ring of bounded (continuous) semialgebraic functions, getting results of a different nature.

In this paper we solve the equation f(g(x))=f(x)hm(x) where f(x), g(x) and h(x) are unknown polynomials with coefficients in an arbitrary field K, f(x) is nonconstant and separable, deg g≥2, the polynomial g(x) has nonzero derivative g′(x)≠0in K[x]and the integer m≥2is not divisible by the characteristic of the field K. We prove that this equation has no solutions if deg f≥3 . If deg f=2 , we prove that m=2and give all solutions explicitly in terms of Chebyshev polynomials. The Diophantine applications for such polynomials f(x) , g(x) , h(x)with coefficients in ℚ or ℤ are considered in the context of the conjecture of Cassaigne et al. on the values of Liouville’s λ function at points f(r) , r∈ℚ.

We refine a result of Dubickas on the maximal multiplicity of the roots of a complex polynomial, and obtain several separability criteria for complex polynomials with large leading coefficient. We also give p-adic analogous results for polynomials with integer coefficients.