triangular decomposition of a bivariate square system by a modular method - Maple Help

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RegularChains[FastArithmeticTools][BivariateModularTriangularize] - triangular decomposition of a bivariate square system by a modular method

Calling Sequence

BivariateModularTriangularize(F, R)

Parameters

R

-

polynomial ring

F

-

bivariate square system of R

Description

• 

The command BivariateModularTriangularize(F, R) returns a triangular decomposition of F in R. See the command Triangularize and the page RegularChains for the concept of a triangular decomposition.

• 

F consists of two bivariate polynomials f1 and f2 of R. No other assumptions are required.

• 

R must have only two variables and no parameters.

• 

Moreover R must have a prime characteristic p such that FFT-based polynomial arithmetic can be used for this actual computation. The higher the degrees of f1 and f2 are, the larger must be e such that 2e divides p1.  If the degree of  f1 or f2 is too large, then an error is raised.

• 

The algorithm is deterministic (i.e. non-probabilistic) and uses modular techniques together with asymptotically fast polynomial arithmetic.

• 

When both Triangularize and BivariateModularTriangularize apply, the latter command is very likely to outperform the former one.

Examples

withRegularChains:

withFastArithmeticTools:

withChainTools:

Define a ring of polynomials.

p:=469762049;vars:=x,y;R:=PolynomialRingvars,p

p:=469762049

vars:=x,y

R:=polynomial_ring

(1)

Define two polynomials of R.

f1:=x49+y+1

f1:=x49+y+1

(2)

f2:=x+y41+1

f2:=y41+x+1

(3)

Compute a triangular decomposition of this system

dec:=BivariateModularTriangularizef1,f2,R

dec:=regular_chain

(4)

mapEquations,dec,R

x+y41+1,y2009+49y1968+1176y1927+18424y1886+211876y1845+1906884y1804+13983816y1763+85900584y1722+450978066y1681+175407438y1640+231867703y1599+10669226y1558+190373232y1517+469560422y1476+200808123y1435+468552287y1394+114869768y1353+112450244y1312+460890461y1271+133871214y1230+200806821y1189+8869201y1148+11288074y1107+319617771y1066+228812073y1025+228812073y984+319617771y943+11288074y902+8869201y861+200806821y820+133871214y779+460890461y738+112450244y697+114869768y656+468552287y615+200808123y574+469560422y533+190373232y492+10669226y451+231867703y410+175407438y369+450978066y328+85900584y287+13983816y246+1906884y205+211876y164+18424y123+1176y82+49y41+469762048y

(5)

Check the number of solutions

mapNumberOfSolutions,dec,R

2009

(6)

See Also

GeneralConstruct, RegularChains, RegularizeDim0, Triangularize


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