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 (1) 
_Env_diffalg_char:
A regular differential system might have no zero. The corresponding components disappear when computing the characteristic decomposition.
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 (2) 
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 (3) 
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 (4) 
Some regular components can split into several characterizable components.
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 (8) 
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 (9) 
The degree in the leaders can decrease.
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 (11) 
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 (12) 
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 (13) 
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 (14) 
_Env_diffalg_charpres_method:
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 (15) 
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 (16) 
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 (17) 
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 (18) 
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 (19) 
The previous example does not succeed using _Env_diffalg_charpres_method := 'groebner'.
_Env_diffalg_charpres_normalize
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 (20) 
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 (21) 
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 (22) 
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 (23) 
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 (24) 
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 (25) 
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 (26) 
_Env_diffalg_primestudy:
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 (27) 
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 (28) 
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 (29) 
This characteristic set is irreducible (the lowest ranked differential polynomial is irreducible and the others have degree one in their leaders). Therefore, the characterizable differential ideal it defines is prime. Thus, the radical differential ideal represented by : is prime. Whatever the ranking, there is a characteristic decomposition of : that contains only one component. Unfortunately a straightforward computation induces some redundant components. They can be eliminated by introducing as a fourth argument to Rosenfeld_Groebner. Setting _Env_diffalg_primestudy:=true can be a good idea to avoid some computations.
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 (30) 
_Env_diffalg_split:
By setting _Env_diffalg_split:= false, one expects to find the most general component. This works in the following example.
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 (31) 
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 (33) 
Nonetheless, setting _Env_diffalg_split to false can lead to misleading results. It might return no component while the input differential polynomials may have a common zero.
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 (34) 
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 (35) 
 (36) 