Introductory examples:
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| (4) |
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| (6) |
The input may contain both radicals and RootOfs, and PseudoDivision will embed the coefficients into an algebraic field, if possible:
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| (7) |
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Nested and mixed radicals and RootOfs are handled as well:
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| (9) |
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Multivariate input is accepted, but pseudo-division will only be performed on the input with respect to the single variable given in the x parameter, with all other names being considered as elements of the coefficient field:
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| (11) |
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| (12) |
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| (13) |
If the degree of a is less than the degree of b, then the multiplier will be and the pseudo-quotient will be :
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If b is a non-zero constant, the pseudo-remainder will always be zero:
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| (15) |
The output will always be fully reduced and normalized:
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| (16) |
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| (17) |
Algebraic Objects will be converted to algebraic numbers, if possible:
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Non-algebraic sub-expressions such as will be frozen and temporarily replaced by new local variables:
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| (19) |
The x parameter can also be a function, as long as it is not an Algebraic Object:
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Arguments of functions in the input will be recursively normalized:
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| (22) |
Non-algebraic sub-expressions may become algebraic after recursive normalization occurs:
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| (23) |
Rational functions are not accepted:
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Floats are not accepted.
Algebraic functions such as are not accepted:
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When non-indexed RootOfs are given in the input, the pseudo-division can still be performed and the output expressed in terms of the non-indexed RootOfs:
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| (24) |
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| (25) |
Even if the leading coefficients of the input contain zero divisors, PseudoDivision can still compute the pseudo-remainder, multiplier, and pseudo-quotient in terms of the input such that the identity is preserved:
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| (26) |
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| (28) |
Also, if the second argument is a zero divisor, then the computation will be performed anyway and an answer will be returned such that the identity is preserved:
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| (29) |
Using option characteristic, pseudo-division can be performed over finite fields:
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| (30) |
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| (32) |
If a RootOf with a non-invertible leading coefficient is detected, an error may be returned:
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In the second case, using option 'symbolic'=true will force PseudoDivision to select one of the factors and perform the computation. Here, it makes the substitution :
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With option 'makeindependent'=true, the input will be checked for algebraic dependencies even if there are more than algebraic objects in the input:
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| (34) |
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| (35) |
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| (36) |
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| (37) |
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| (38) |
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| (39) |
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| (40) |
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With option 'makeindependent'=false, the input will never be checked for algebraic dependencies:
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| (43) |