Reduced Length and Mahler-Measure Inequality

Definitions and mathematical statement

For $R(x)=\sum_{j=0}^r r_jx^j\in{\mathbb{R}}[x]$, define its length and reduced length by \[ L(R)=\sum_{j=0}^r|r_j|,\qquad \ell(R)=\inf_{\substack{Q\in{\mathbb{R}}[x]\\Q\text{ monic}}}L(RQ). \] The infimum ranges over all degrees of monic real polynomials, including the constant polynomial $1$; no minimizing $Q$ is assumed to exist.

For a nonzero polynomial $P(x)=a\prod_{j=1}^m(x-\alpha_j)$ with roots counted with multiplicity, its multiplicative Mahler measure is \[ M(P)=|a|\prod_{j=1}^m\max\{1,|\alpha_j|\}. \] For a nonzero constant this is its absolute value. The zero polynomial can be included by the usual extensions $M(0)=0$ and $\ell(0)=0$; otherwise it may be treated separately.

The conjectural inequality is \[ \forall t\in[-2,2]\ \forall P\in{\mathbb{R}}[x]:\qquad \ell\left((x^2+tx+1)P(x)\right)\ge2M(P). \] Equivalently, with the same $t,P$ quantifiers, every monic $Q\in{\mathbb{R}}[x]$ should satisfy \[ L\left((x^2+tx+1)P(x)Q(x)\right)\ge2M(P). \] The Mahler measure on the right is that of $P$, not of a different product and not its logarithm. There is no monicity or integrality requirement on $P$. Both endpoints $t=-2,2$ are included.

The supplied literature assessment describes the question as posed for all real $t$, whereas both original and restored statements explicitly restrict $t$ to $[-2,2]$. This discrepancy is retained: the draft's mathematical target is the supplied interval statement and is not silently broadened to every real parameter.

Short English statement

For every real polynomial and every $t$ in the closed interval $[-2,2]$, is the reduced coefficient length of $(x^2+tx+1)P$ at least twice the multiplicative Mahler measure of $P$? The multiplier infimum includes monic polynomials of arbitrary degree.

Sources

• Local UnsolvedMath registry, permanent ID 30000750, code catalogue entry, "Reduced Length and Mahler-Measure Inequality." Proposed year 2007; proposer unspecified.
• Diophantische Approximationen (2008), Oberwolfach Reports, pp. 1115--1190, DOI 10.4171/owr/2007/21; https://doi.org/10.4171/owr/2007/21. The displayed report year 2008 differs from the DOI and proposed year 2007. This is the sole supplied literature reference.
• Statement status corrected_verified; the supplied note says the complete inequality was retained and infimum, ring, exponent, and interval notation restored.
• Literature triage checked 2026-08-21: open. It reports no definitive proof or counterexample located, but describes "all real $t$" instead of the explicit interval in the statement. That scope discrepancy and current status require independent verification. Created 2026-08-25; updated 2026-09-16.
• Registry curation by Ulam AI / UnsolvedMath Contributors, CC BY 4.0; the underlying report retains its supplied provenance. Unreviewed statement-only draft.