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<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\mathbb {F}_q[t]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> be the polynomial ring over the finite field <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathbb {F}_q\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> </math></EquationSource> </InlineEquation> of <i>q</i> elements. A polynomial in <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\mathbb {F}_q[t]\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi mathvariant="double-struck">F</mi> <mi>q</mi> </msub> <mrow> <mo stretchy="false">[</mo> <mi>t</mi> <mo stretchy="false">]</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> is called <i>m</i>-smooth (or <i>m</i>-friable) if all its irreducible factors are of degree at most <i>m</i>. In this paper, we investigate the distribution of <i>m</i>-smooth (or <i>m</i>-friable) polynomials with prescribed coefficients. Our technique is based on character sum estimates on smooth (friable) polynomials, Bourgains’s argument (2015) applied for polynomials by Ha (2016) and on double character sums on smooth (friable) polynomials.</p>

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Smooth polynomials with several prescribed coefficients

  • 期刊论文

  • 发表日期：

    2026年6月15日

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  • László Mérai

摘要

Let \(\mathbb {F}_q[t]\) F q [ t ] be the polynomial ring over the finite field \(\mathbb {F}_q\) F q of q elements. A polynomial in \(\mathbb {F}_q[t]\) F q [ t ] is called m-smooth (or m-friable) if all its irreducible factors are of degree at most m. In this paper, we investigate the distribution of m-smooth (or m-friable) polynomials with prescribed coefficients. Our technique is based on character sum estimates on smooth (friable) polynomials, Bourgains’s argument (2015) applied for polynomials by Ha (2016) and on double character sums on smooth (friable) polynomials.