where
is the usual complex multiplication. The
two factors are chosen independently. The operation is
commutative and associative,
but in general it satisfies only , where denotes the Minkowski
sum (Farouki et al. 2001, pp. 284-285). For example, if and , then , while .
Indeed, ,
and for fixed
the possible products satisfy . The points and belong to the product, but their midpoint
does not. The image is only the part of the parabola with .
Farouki, R. T.; Gu, W.; and Moon, H. P. "Minkowski Roots of Complex Sets." In Proc. Geometric Modeling and Processing 2000:
Theory and Applications (Ed. R. Martin and W. Wang). Los Alamitos,
CA: IEEE Computer Society Press, pp. 287-300, 2000. https://doi.org/10.1109/GMAP.2000.838260.Farouki,
R. T. and Han, C. Y. "Computation of Minkowski Values of Polynomials
over Complex Sets." Numer. Algorithms36, 13-29, 2004. https://doi.org/10.1023/B:NUMA.0000027725.80448.d8.Farouki,
R. T. and Moon, H. P. "Minkowski Geometric Algebra and the Stability
of Characteristic Polynomials." In Visualization and Mathematics III
(Ed. H.-C. Hege and K. Polthier). Berlin, Germany: Springer-Verlag, pp. 163-188,
2003. https://doi.org/10.1007/978-3-662-05105-4_9.Farouki,
R. T.; Moon, H. P.; and Ravani, B. "Algorithms for Minkowski Products
and Implicitly-Defined Complex Sets." Adv. Comput. Math.13, 199-229,
2000. https://doi.org/10.1023/A:1018910412112.Farouki,
R. T.; Moon, H. P.; and Ravani, B. "Minkowski Geometric Algebra of
Complex Sets." Geom. Dedicata85, 283-315, 2001. https://doi.org/10.1023/A:1010318011860.Trott,
M. "Polynomials in Inequalities." §1.2.3 in The
Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5.
https://www.mathematicaguidebooks.org/.
is the set
is the usual complex multiplication. The
two factors are chosen independently. The operation is
commutative and associative,
but in general it satisfies only 
, where 
denotes the Minkowski
sum (Farouki et al. 2001, pp. 284-285). For example, if 
and 
, then 
, while 
.
,
,
and for fixed 
the possible products satisfy 
. The points 
and 
belong to the product, but their midpoint
does not. The image 
is only the part of the parabola 
with 
.
and 
gives the real equations 
and 
for 
. If 
and 
are semialgebraic sets,
these equations express membership in 
using existential
quantifiers. Quantifier elimination
therefore gives a semialgebraic set description
of the product (Trott 2006).
