The simple-roots conjecture for finite difference operators

From papers

Let $p$ be a complex polynomial, and let $\Delta_{\theta,h}$ be the finite difference operator from the paper. The exceptional polynomials are those with multiple roots of the form $\pm nih$ , where $n\in\mathbb{N}$ .

Simple-roots conjecture. For almost all polynomials $p$ , the polynomial $\Delta_{\theta,h}(p)$ has only simple roots, apart from the stated exclusion.

The conjecture is motivated by the observation that most polynomials in the image of the operator have simple roots, while the source describes the exceptional multiple-root configurations. No resolution is given.

Progress summary

Open

The conjecture remains unresolved: it predicts that repeated roots are rare except for listed special patterns, but gives no proof.

The paper records Conjecture $3$ : for almost all complex polynomials, the associated finite-difference output has only simple roots, apart from explicitly described configurations involving points of the form $\pm nih$ , with $n\in\mathbb{N}$ . It presents the conjecture as motivated by computations and gives no proof or resolution.

Current status (as of August 2026): the conjecture remains open, with no public proof, counterexample, or verification recorded in the retrieved sources.

Sources

ar5iv.labs.arxiv.org

Sources & referencesView supporting material

Primary source

Olga Katkova, Mikhail Tyaglov and Anna Vishnyakova, “Hermite-Poulain theorems for linear finite difference operators”, arXiv:1901.06398 (2019).

finite-difference-operators polynomial-roots

1 week ago

The generic simple-root assertion is true, but the proposed exhaustive characterization of its exceptions is false.

Fix $h\ne0$ , $\theta\in[0,\pi)$ , and the input degree $d$ . The condition that

\Delta_{\theta,h}p(z) =\frac{e^{i\theta}p(z+ih)-e^{-i\theta}p(z-ih)}{2i}

have a multiple zero is the vanishing of the discriminant of the output polynomial, a polynomial in the coefficients of $p$ . This discriminant is not identically zero: for $p(z)=z^d$ , the finite zeros are determined by

\left(\frac{z+ih}{z-ih}\right)^d=e^{-2i\theta},

and are distinct. When $\theta=0$ , the ratio $1$ corresponds to the degree drop and the remaining $d-1$ finite zeros are distinct. Thus the exceptional locus is a proper algebraic hypersurface, establishing the generic assertion.

However, the exceptional locus is not restricted to the multiple-root inputs specified in the conjecture. Take

\theta=0,\qquad h=1,\qquad p(z)=z^3+z=z(z-i)(z+i).

This input has three distinct zeros, so it belongs to none of the claimed exceptional classes. Nevertheless,

\Delta_{0,1}p(z) =\frac{p(z+i)-p(z-i)}{2i} =3z^2,

which has a double zero at $z=0$ .

Therefore the generic statement holds, while the proposed characterization of all exceptional polynomials is disproved.

0 endorsements

Shivam Patel · 1 day ago