| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>(</mo><mi>V</mi><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>=</mo><mo>{</mo><mrow><mo>(</mo><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>z</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>∈</mo><msup><mi mathvariant="double-struck">C</mi><mi>n</mi></msup><mspace width="4pt"></mspace><mo>:</mo><mspace width="4pt"></mspace><mi>f</mi><mrow><mo>(</mo><msub><mi>z</mi><mn>1</mn></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mi>z</mi><mi>n</mi></msub><mo>)</mo></mrow><mo>=</mo><mn>0</mn><mo>}</mo></mrow></semantics></math></inline-formula> be an isolated hypersurface singularity with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>m</mi><mi>u</mi><mi>l</mi><mi>t</mi><mo>(</mo><mi>f</mi><mo>)</mo><mo>=</mo><mi>m</mi></mrow></semantics></math></inline-formula>. Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>J</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> be the ideal generated by all <i>k</i>-th order partial derivatives of <i>f</i>. For <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>k</mi><mo>≤</mo><mi>m</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, the new object <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">L</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is defined to be the Lie algebra of derivations of the new <i>k</i>-th local algebra <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>M</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msub><mi mathvariant="script">O</mi><mi>n</mi></msub><mo>/</mo><mrow><mo>(</mo><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>+</mo><msub><mi>J</mi><mn>1</mn></msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>+</mo><mo>…</mo><mo>+</mo><msub><mi>J</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>f</mi><mo>)</mo></mrow><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. Its dimension is denoted as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. This number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> is a new numerical analytic invariant. In this article we compute <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="script">L</mi><mn>4</mn></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for fewnomial isolated singularities (binomial, trinomial) and obtain the formulas of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mn>4</mn></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>. We also verify a sharp upper estimate conjecture for the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mn>4</mn></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula> for large class of singularities. Furthermore, we verify another inequality conjecture: <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>δ</mi><mrow><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo><</mo><msub><mi>δ</mi><mi>k</mi></msub><mrow><mo>(</mo><mi>V</mi><mo>)</mo></mrow><mo>,</mo><mspace width="0.166667em"></mspace><mi>k</mi><mo>=</mo><mn>3</mn></mrow></semantics></math></inline-formula> for low-dimensional fewnomial singularities.
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