| Summary: | <p>It is easy to see that an element <em>P</em>(<em>t</em>)∈<strong>F</strong><sub>2</sub>[<em>t</em>] is a sum of cubes if and only if</p> <p align="center"><em>P</em>(<em>t</em>)∈<em>M</em>(2):<strong>=</strong>{<em>P</em>(<em>t</em>): <em>P</em>(<em>t</em>)≡ 0 or 1 (mod <em>t</em><sup>2</sup>+<em>t</em>+1)}.</p><p>We say that <em>P</em>(<em>t</em>) is a "strict" sum of cubes <em>A</em><sub>1</sub>(<em>t</em>)<sup>3</sup>+<sup>...</sup>+ <em>A</em><sub><em>g</em></sub>(<em>t</em>)<sup>3</sup> if we have deg (<em>A</em><sup>3</sup><substyle='position: -.8em;'="" left:="" relative;="">i</substyle='position:></p>) ≤ deg (<em>P</em>)+2 for each <em>i</em>, and we define <em>g</em>(3,<strong>F</strong><sub>2</sub>[<em>t</em>]) as the least <em>g</em> such that every element of <em>M</em>(2) is a strict sum of <em>g</em> cubes. Our main result is then that<p align="center">5≤<em>g</em>(3,<strong>F</strong><sub>2</sub>[<em>t</em>])≤6.</p> <p>This improves on a recent result 4≤<em>g</em>(3,<strong>F</strong><sub>2</sub>[<em>t</em>])≤9 of the first named author. © 2007 Elsevier Inc. All rights reserved.</p>
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