How do you calculate the number coefficients in polynomial regression?
Yes - it is $d\times l$. There would be 6 $β$ coefficients for 2*3:
$β_1x_1 + β_2x_1^2 + β_3x_2 + β_4x_2^2 + β_5x_3 + β_6x_3^2$
That does not include an intercept or any interaction terms.