Type: Article
Publication Date: 1995-01-01
Citations: 391
DOI: https://doi.org/10.1090/s0273-0979-1995-00571-9
We provide an elementary geometric derivation of the Kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients. We show that the expected number of real zeros is simply the length of the moment curve <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="left-parenthesis 1 comma t comma ellipsis comma t Superscript n Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mn>1</mml:mn> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mi>t</mml:mi> <mml:mo>,</mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo>…<!-- … --></mml:mo> <mml:mspace width="thinmathspace" /> <mml:mo>,</mml:mo> <mml:msup> <mml:mi>t</mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">(1,\,t,\,\ldots \,,t^{n})</mml:annotation> </mml:semantics> </mml:math> </inline-formula> projected onto the surface of the unit sphere, divided by <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="pi"> <mml:semantics> <mml:mi>π<!-- π --></mml:mi> <mml:annotation encoding="application/x-tex">\pi</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. The probability density of the real zeros is proportional to how fast this curve is traced out. We then relax Kac’s assumptions by considering a variety of random sums, series, and distributions, and we also illustrate such ideas as integral geometry and the Fubini-Study metric.