Editing Bateman–Horn conjecture (section)

==Analogue for polynomials over a finite field==

When the integers are replaced by the polynomial ring ''F''[''u''] for a finite field ''F'', one can ask how often a finite set of polynomials ''f''<sub>''i''</sub>(''x'') in ''F''[''u''][''x''] simultaneously takes  irreducible values in ''F''[''u''] when we substitute for ''x'' elements of ''F''[''u''].  Well-known analogies between integers and ''F''[''u''] suggest an analogue of the Bateman–Horn conjecture over ''F''[''u''], but the analogue is wrong.  For example, data suggest that the polynomial

::<math>x^3 + u\,</math>

in ''F''<sub>3</sub>[''u''][''x''] takes (asymptotically) the expected number of irreducible values when ''x'' runs over polynomials in ''F''<sub>3</sub>[''u''] of odd degree, but it appears to take (asymptotically) twice as many irreducible values as expected when ''x'' runs over polynomials of degree that is 2 mod 4, while it (provably) takes ''no'' irreducible values at all when ''x'' runs over nonconstant polynomials with degree that is a multiple of 4. An analogue of the Bateman–Horn conjecture over ''F''[''u''] which fits numerical data uses an additional factor in the asymptotics which depends on the value of ''d'' mod 4, where ''d'' is the degree of the polynomials in ''F''[''u''] over which ''x'' is sampled.