Co.Al.A.R.
Computational Algebraic Analysis Results
 

THIS PAGE IS DEVOTED TO SOME RESULTS IN CLIFFORD ANALYSIS and related topics


last update: January, 5th 2006

for any question on this site or any of the CoAlA webpages, please contact adamiano@gmu.edu

 
==>  NOETHERIAN OPERATORS [DSS]:
 
 Some experiments performed using different algorithms, includes CPU times
 
 variables 
multiplicity IDEAL CPU TIME (CoCoA 4.5 on Toshiba Sat. 2455)

- 3.4 from [DSS], following an idea of [MMM]

- (40) from [Ob96], using linear algebra

- 3.8  from [DSS], using forward reduction

- 3.17 from [DSS], using backward reduction

OPERATORS
2
4
(x^2-y ,y^2)
0.75''

0.14''

0.37''

0.08''
                                       

1 , 
dx , 
dx^2+dy ,
dx^3+dxdy
2
8
(x^4-xy-y, y^2)
8.31''

0.90''

0.28 ''

0.15''

1 ,
dx ,
dx^2 ,
dx^3 ,
dx^4+dy ,
dx^5+dx^4+dxdy ,
dx^6+dx^5+dx^2dy ,
dx^7+dx^6+dx^3dy
2
9
(x^3-y, y^3)
2' 9''

1.53''

0.34''

0.22''

1 ,
dx ,
dx^2 ,
dx^3+dy ,
dx^4+dxdy ,
dx^5+dx^2dy ,
dx^6+dx^3dy+dy^2 ,
dx^7+dx^4dy+dxdy^2 ,
dx^8+dx^5dy+dx^2dy^2
3
8
(x^2-z, y^2-z, z^2)
                             
12'

9.18''

0.99''

0.19''

1 ,
dy ,
dx ,
dxdy ,
dx^2+dy^2+dz ,
dx^2dy+dy^3+dydz ,
dx^3+dxdy^2+dxdz ,
dx^3dy+dxdy^3+dxdydz
3
4
(x^2-ty ,y^2) dim(I)>0

dim(I)>0

0.64''

not yet available

1 ,
dx ,
tdx^2+dy ,
tdx^3+dxdy
4
8
(x^4-txy-sy, y^2)
dim(I)>0

dim(I)>0

2.61''

not yet available

1 ,
dx ,
dx^2 ,
dx^3 ,
sdx^4+dy ,
sdx^5+tdx^4+dxdy ,
sdx^6+tdx^5+dx^2dy ,
sdx^7+tdx^6+dx^3dy
5
8
(x^2-tz, y^2-sz, z^2)
                                 
dim(I)>0

dim(I)>0

12.73''

not yet available

1 ,
dy ,
dx ,
dxdy ,
sdy^2+tdx^2+dz ,
sdy^3+tdx^2dy+dydz ,
sdxdy^2+tdx^3+dxdz ,
sdxdy^3+tdx^3dy+dxdydz
 

 
 
 
REFERENCES: (click on [XXX] to get the AMS reference #)


[DSS]
A. Damiano, I. Sabadini, D. Struppa,  Computational Methods for the Construction of a Class of Noetherian Operators, Submitted
 
[Ob96] U. OberstFinite-dimensional systems of partial differential or difference equations. Adv. in Appl. Math. 17 (1996), no. 3, 337--356.
 
[MMM] Marinari, M. G.; Möller, H. M.; Mora, T. On multiplicities in polynomial system solving. Trans. Amer. Math. Soc. 348 (1996), no. 8, 3283--3321.