Basic Element Operations

Parent and Category

Parent(r): RngSerElt -> RngSer

Category(r): RngSerElt -> Cat

Arithmetic Operators

+ b: RngSerElt -> RngSerElt

- b: RngSerElt -> RngSerElt

a + b: RngSerElt, RngSerElt -> RngSerElt

a - b: RngSerElt, RngSerElt -> RngSerElt

a * b: RngSerElt, RngSerElt -> RngSerElt

a ^ k: RngSerElt, RngIntElt -> RngSerElt

a div b: RngSerPowElt, RngSerPowElt -> RngSerPowElt

a / b: RngSerElt, RngSerElt -> RngSerElt

Equality and Membership

a eq b: RngSerElt, RngSerElt -> BoolElt

a ne b: RngSerElt, RngSerElt -> BoolElt

a in R: RngSerElt, RngSer -> BoolElt

a notin R: RngSerElt, RngSer -> BoolElt

Predicates on Ring Elements

Note the definition of equality in the introduction to this Chapter. This not only affects the result of the application of eq and ne, but also that of IsOne, IsZero and IsMinusOne.

IsZero(a): RngSerElt -> BoolElt
IsOne(a): RngSerElt -> BoolElt
IsMinusOne(a): RngSerElt -> BoolElt
IsNilpotent(x): RngSerElt -> BoolElt
IsIdempotent(x): RngSerElt -> BoolElt
IsUnit(a): RngSerElt -> BoolElt
IsZeroDivisor(x): RngSerElt -> BoolElt
IsRegular(x): RngSerElt -> BoolElt
IsIrreducible(x): RngSerElt -> BoolElt
IsPrime(x): RngSerElt -> BoolElt
IsWeaklyZero(f): RngSerElt -> BoolElt: Given a series \(f\), return whether \(f\) is weakly zero, which is whether \(f\) is exactly zero or of the form \(O(x^p)\) for some \(p\).

IsWeaklyEqual(f, g): RngSerElt, RngSerElt -> BoolElt: Given series \(f\) and \(g\), return whether \(f\) is weakly equal to \(g\), which is whether (\(f-g\)) is weakly zero (see IsWeaklyZero).

IsIdentical(f, g): RngSerElt, RngSerElt -> BoolElt: Given series \(f\) and \(g\), return whether \(f\) is identical to \(g\), which is whether \(f\) and \(g\) have exactly the same valuation, precision, and coefficients.

Precision

AbsolutePrecision(f): RngSerElt -> RngIntElt: Given a series \(f\), this returns the absolute precision that is stored with \(f\). If \(f\) is a series in \(x\), the absolute precision of \(f\) is the exponent \(p\) such that \(x^p\) is the first term of \(f\) of which the coefficient is not known, that is, it is the least \(p\) such that \(f\in O(x^p)\). If \(f\) is known exactly (in a free ring), the absolute precision is infinite and an error occurs. Note that the absolute precision may be a non-integral rational number if \(f\) is a Puiseux series.

RelativePrecision(f): RngSerElt -> RngIntElt: Given a series \(f\), this returns the relative precision that is stored with \(f\). The relative precision counts the number of coefficients of \(f\) that is known, starting at the first non-zero term. Hence the relative precision is the difference between the absolute precision and the valuation of \(f\), and is therefore always non-negative; however, if \(f\) is exact, the relative precision is infinite and the value \(\infty\) is returned. Note that the relative precision may be a non-integral rational number if \(f\) is a Puiseux series.

ChangePrecision(f, r): RngSerElt, RngIntElt -> RngSerElt
ChangePrecision(f, r): RngSerElt, Infty -> RngSerElt
ChangePrecision(~f, r): RngSerElt, RngIntElt
ChangePrecision(~f, r): RngSerElt, Infty: The (non puiseux) series \(f\) with absolute precision \(r\) (which can be positive infinity).

Coefficients and Degree

Coefficients(f): RngSerElt -> [ RngElt ], RngIntElt, RngIntElt
ElementToSequence(f): RngSerElt -> [ RngElt ], RngIntElt, RngIntElt
Eltseq(f): RngSerElt -> [ RngElt ], RngIntElt, RngIntElt: Let \(f\) be a series with coefficients in a ring \(R\) and with indeterminate \(x\). This function returns the sequence \(Q\) of coefficients of \(f\), the unscaled valuation \(v\), and the exponent denominator \(d\) of \(f\) (\(v\) is the true valuation of \(f\) multiplied by \(d\)). The \(i\)-th entry \(Q[i]\) of \(Q\) equals the coefficient of \(x^{{v+i-1}\over{d}}\) in \(f\). Thus the first entry of \(Q\) is the ‘first’ (lowest order) non-zero coefficient of \(f\), i.e., the coefficient of \(x^w\) where \(w\) is the true valuation of \(f\).

Coefficient(f, i): RngSerElt, RngElt -> RngElt: Given a series \(f\) with coefficients in a ring \(R\), and a rational or integer \(i\), return the coefficient of the \(i\)-th power of the indeterminate \(x\) of \(f\) as an element of \(R\). If \(f\) is a Puiseux series \(i\) may be a (non-integral) rational; otherwise \(i\) must be an integer (and also must be non-negative if \(f\) is a power series). Also, \(i\) must be less than \(p\), the precision of \(f\).

LeadingCoefficient(f): RngSerElt -> RngElt: Given a series \(f\) with coefficients in a ring \(R\), return the leading coefficient of \(f\) as an element of \(R\), which is the first non-zero coefficient of \(f\) (i.e., the coefficient \(x^v\) in \(f\), where \(x\) is the indeterminate of \(f\) and \(v\) is the valuation of \(f\)).

LeadingTerm(f): RngSerElt -> RngElt: Given a series \(f\) with coefficients in a ring \(R\), return the leading term of \(f\), which is the first non-zero term of \(f\) (i.e., the term of \(f\) whose monomial is \(x^v\), where \(x\) is the indeterminate of \(f\) and \(v\) is the valuation of \(f\)).

Truncate(f): RngSerElt -> RngSerElt: Given a series \(f\), return the exact series obtained by truncating \(f\) after the last known non-zero coefficient.

ExponentDenominator(f): RngMSerElt -> RngElt: Given a series \(f\), return the exponent denominator of \(f\), i.e., the lowest common denominator of all the exponents of the non-zero terms of \(f\) (always an integer). For power series and Laurent series, this will always be \(1\) of course.

Degree(f): RngSerElt -> RngIntElt: Given a series \(f\), return the degree of the truncation of \(f\), that is, the exponent of the last known non-zero term. Note that this may be a non-integral rational number if \(f\) is a Puiseux series.

Valuation(f): RngSerElt -> RngIntElt: Given a series \(f\), return the smallest integer \(v\) (possibly negative for Laurent series) such that the coefficient of \(x^v\) in \(f\) is not known to be zero. For the exact \(0\) element (in a free ring), the valuation is \(\infty\). Note that the valuation may be a non-integral rational number if \(f\) is a Puiseux series.

ExponentDenominator(f): RngSerElt -> RngIntElt: The exponent denominator of the series \(f\). This is the lowest common denominator of the exponents of the non-zero terms of \(f\).

Evaluation and Derivative

Derivative(f): RngSerElt -> RngSerElt: Given a series \(f\in R\), return the derivative of \(f\) with respect to its indeterminate, as an element of \(R\). Note that the precision decreases by 1 (unless \(f\) has infinite precision).

Derivative(f, n): RngSerElt, RngIntElt -> RngSerElt: Given a series \(f\in R\) and an integer \(n > 0\), return the \(n\)-th derivative of \(f\) with respect to its indeterminate, as an element of \(R\). Note that the precision decreases by \(n\) (unless \(f\) has infinite precision).

Integral(f): RngSerElt -> RngSerElt: Given a series \(f\in R\), return an anti-derivative \(F\) of \(f\) with respect to its indeterminate, which is an element of \(R\) which has derivative \(f\). The coefficient of \(x^{-1}\) in \(f\) must be zero. Note that the precision of \(F\) will be exceeding that of \(f\) by 1 (unless \(f\) has infinite precision).

Evaluate(f, s): RngSerElt, RngElt -> RngElt: Given an element \(f\) of a series ring over the coefficient ring \(R\), and an element \(s\) of the ring \(S\), return the value of \(f(s)\) when the indeterminate \(x\) is evaluated at \(s\). The result will be an element of the common overstructure over \(R\) and \(S\).

Laplace(f): RngSerElt -> RngSerElt: The Laplace transform of the series \(f\); if \(f\) has expansion \(\sum_{i\geq 0} a_ix^i\), its Laplace transform has expansion \(\sum_{i\geq 0} (i!a_i)x^i\). The valuation of \(f\) must be integral and non-negative.

Square Root

SquareRoot(f): RngSerElt -> RngSerElt
Sqrt(f): RngSerElt -> RngSerElt: Return the square root of the series \(f\), \(f\) must have even valuation if it is a power or Laurent series.

Composition and Reversion

Composition(f, g): RngSerElt, RngSerElt -> RngSerElt

Given elements \(f\) and \(g\) from the same series ring \(P\), return their composition, defined by

\[f\circ g = \sum_{i < p}f_i(g^i),\]

where \(f=\sum_{i < p} f_ix^i\).

Reversion(f): RngSerElt -> RngSerElt
Reverse(f): RngSerElt -> RngSerElt: Given a series \(f\) (in \(x\), say), this returns the inverse of \(f\) under composition, that is, an element \(g\) of the same power series ring such that its composition with \(f\) equals \(x\) to the best possible precision. If \(f\) is a power or Laurent series, the valuation of \(f\) must be 1. If \(f\) is a Puiseux series, the valuation of \(f\) must be positive (but need not equal 1), and if the valuation of \(f\) is not 1, the leading coefficient of \(f\) must be 1.

Convolution(f, g): RngSerElt, RngSerElt -> RngSerElt

Given elements \(f\) and \(g\) from the same series ring \(P\), return their convolution \(f\ast g\), defined by

\[f\ast g = \sum_{i < \min(p,q)}f_ig_ix^i,\]

where \(f=\sum_{i < p} f_ix^i+O(x^p)\) and \(g=\sum_{i < q}g_ix^i+O(x^q)\).

Example: Composition and Reversion of Series (ex-cad173)

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We demonstrate the functions Composition and Reversion. First we check that Arcsin is the reversion of Sin.

> S<x> := PowerSeriesRing(RationalField());
> f := Sin(x);
> g := Arcsin(x);
> f;
x - 1/6*x^3 + 1/120*x^5 - 1/5040*x^7 + 1/362880*x^9 -
    1/39916800*x^11 + 1/6227020800*x^13 - 1/1307674368000*x^15 +
    1/355687428096000*x^17 - 1/121645100408832000*x^19 + O(x^21)
> g;
x + 1/6*x^3 + 3/40*x^5 + 5/112*x^7 + 35/1152*x^9 + 63/2816*x^11 +
    231/13312*x^13 + 143/10240*x^15 + 6435/557056*x^17 +
    12155/1245184*x^19 + O(x^21)
> Composition(f, g);
x + O(x^21)
> Composition(g, f);
x + O(x^21)
> Reversion(f) - g;
O(x^21)
> Reversion(g) - f;
O(x^21)

Next we compute the reversion of a series whose valuation is not 1.

> S<x> := PuiseuxSeriesRing(RationalField());
> f := x^3 - x^5 + 2*x^8;
> r := Reversion(f);
> f;
x^3 - x^5 + 2*x^8
> r;
x^(1/3) + 1/3*x + 4/9*x^(5/3) - 2/3*x^2 + 65/81*x^(7/3) -
    22/9*x^(8/3) + 5/3*x^3 - 208/27*x^(10/3) + 5005/729*x^(11/3)
    - 70/3*x^4 + 206264/6561*x^(13/3) - 50830/729*x^(14/3) +
    134*x^5 - 498674/2187*x^(16/3) + 31389020/59049*x^(17/3) +
    O(x^6)
> Composition(r, f);
x + O(x^18)
> Composition(f, r);
x + O(x^(20/3))

Finally we compute the reversion of a proper Puiseux series.

> f := x^(2/5) - x^(2/3) + x^(3/2) + O(x^2);
> r := Reversion(f);
> r;
x^(5/2) + 5/2*x^(19/6) + 145/24*x^(23/6) + 715/48*x^(9/2) +
    389795/10368*x^(31/6) - 5/2*x^(21/4) + O(x^(11/2))
> Composition(f, r);
x + O(x^4)
> Composition(r, f);
x + O(x^(11/5))

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Basic Element Operations#

Parent and Category#

Arithmetic Operators#

Equality and Membership#

Predicates on Ring Elements#

Precision#

Coefficients and Degree#

Evaluation and Derivative#

Square Root#

Composition and Reversion#