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- W3190387617 abstract "Abstract We give an explicit formula for the expectation of the number of real lines on a random invariant cubic surface, i.e., a surface $$Zsubset {mathbb {R}}{mathrm {P}}^3$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Z</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>R</mml:mi> <mml:msup> <mml:mrow> <mml:mi>P</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> </mml:mrow> </mml:math> defined by a random gaussian polynomial whose probability distribution is invariant under the action of the orthogonal group O (4) by change of variables. Such invariant distributions are completely described by one parameter $$lambda in [0,1]$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>[</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>1</mml:mn> <mml:mo>]</mml:mo> </mml:mrow> </mml:math> and as a function of this parameter the expected number of real lines equals: $$begin{aligned} E_lambda =frac{9(8lambda ^2+(1-lambda )^2)}{2lambda ^2+(1-lambda )^2}left( frac{2lambda ^2}{8lambda ^2+(1-lambda )^2}-frac{1}{3}+frac{2}{3}sqrt{frac{8lambda ^2+(1-lambda )^2}{20lambda ^2+(1-lambda )^2}}right) . end{aligned}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mi>λ</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mrow> <mml:mn>9</mml:mn> <mml:mo>(</mml:mo> <mml:mn>8</mml:mn> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mfenced> <mml:mfrac> <mml:mrow> <mml:mn>2</mml:mn> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mn>8</mml:mn> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> <mml:mo>-</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:mo>+</mml:mo> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> <mml:msqrt> <mml:mfrac> <mml:mrow> <mml:mn>8</mml:mn> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> <mml:mrow> <mml:mn>20</mml:mn> <mml:msup> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>+</mml:mo> <mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>-</mml:mo> <mml:mi>λ</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:mrow> </mml:mfrac> </mml:msqrt> </mml:mfenced> <mml:mo>.</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> This result generalizes previous results by Basu et al. (Math Ann 374(3–4):1773–1810, 2019) for the case of a Kostlan polynomial, which corresponds to $$lambda =frac{1}{3}$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:mrow> </mml:math> and for which $$E_{frac{1}{3}}=6sqrt{2}-3.$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:mfrac> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>6</mml:mn> <mml:msqrt> <mml:mn>2</mml:mn> </mml:msqrt> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> <mml:mo>.</mml:mo> </mml:mrow> </mml:math> Moreover, we show that the expectation of the number of real lines is maximized by random purely harmonic cubic polynomials, which corresponds to the case $$lambda =1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>=</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> and for which $$E_1=24sqrt{frac{2}{5}}-3$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>E</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>24</mml:mn> <mml:msqrt> <mml:mfrac> <mml:mn>2</mml:mn> <mml:mn>5</mml:mn> </mml:mfrac> </mml:msqrt> <mml:mo>-</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> ." @default.
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- W3190387617 date "2021-07-02" @default.
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- W3190387617 title "Real Lines on Random Cubic Surfaces" @default.
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