| Summary: | We study equations of the form <inline-formula> <math display="inline"> <semantics> <mrow> <mi>y</mi> <mi>d</mi> <mi>y</mi> <mo>/</mo> <mi>d</mi> <mi>x</mi> <mo>=</mo> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> where <inline-formula> <math display="inline"> <semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>)</mo> <mo>∈</mo> <mspace width="3.33333pt"></mspace> <mi mathvariant="double-struck">R</mi> <mo>[</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>]</mo> </mrow> </semantics> </math> </inline-formula> with degree <i>n</i> in the <i>y</i>-variable. We prove that this ordinary differential equation has at most <i>n</i> polynomial solutions (not necessarily constant but coprime among each other) and this bound is sharp. We also consider polynomial limit cycles and their multiplicity.
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