Multivariate polynomials

Contents

• Creation of rings
• Basic operations on rings
• Creation of polynomials
  • Coercion
• Basic operations on polynomials
  • Derivative
  • Evaluate
• Hensel lifting
• Internals
  • Approximation

Creation of rings

PolynomialRing (F :: FldPadExact, k :: RngIntElt)

-> RngMPol_FldPadExact

The polynomial ring of rank k over F.

Basic operations on rings

BaseRing (R :: RngMPol_FldPadExact)

-> FldPadExact

The base ring of R.

Rank (R :: RngMPol_FldPadExact)

-> RngIntElt

The rank of R.

Generator (R :: RngMPol_FldPadExact, i :: RngIntElt)

-> RngMPolElt_FldPadExact

The ith generator of R.

Generators (R :: RngMPol_FldPadExact)

-> []

The generators of R.

Creation of polynomials

Coercion

The following are coercible to a multivariate polynomial in R:

• A polynomial in R.
• A multivariate polynomial of correct rank whose coefficients are coercible into the base ring of R.
• A sequence of tuples, whose first element is a coefficient and whose second element is an exponent vector.
• Anything coercible to the base ring of R.

IsCoercible (R :: RngMPol_FldPadExact, X)

-> BoolElt, Any

True if X is coercible to an element of R. If so, also returns the coerced element.

Basic operations on polynomials

MonomialCoefficient (f :: RngMPolElt_FldPadExact, m :: RngMPolElt_FldPadExact)

-> FldPadExactElt

The coefficient of monomial m in f.

ExponentsCoefficient (f :: RngMPolElt_FldPadExact, e :: [RngIntElt])

-> FldPadExactElt

The coefficient of exponent e in f.

Monomial (R :: RngMPol_FldPadExact, e :: [RngIntElt])

-> FldPadExactElt

The monomial of R with exponents e.

IsDefinitelyMonomial (f :: RngMPolElt_FldPadExact)

-> BoolElt, []

True if f is definitely a monomial (i.e. has one term). If so, also returns its exponents.

Exponents (m :: RngMPolElt_FldPadExact)

-> []

The exponents of m, which must be a monomial.

'-' (f :: RngMPolElt_FldPadExact)

'+' (f :: RngMPolElt_FldPadExact, g :: RngMPolElt_FldPadExact)

'-' (f :: RngMPolElt_FldPadExact, g :: RngMPolElt_FldPadExact)

'*' (f :: RngMPolElt_FldPadExact, g :: RngMPolElt_FldPadExact)

'/' (f :: RngMPolElt_FldPadExact, x :: FldPadExactElt)

'&+' (fs :: [RngMPolElt_FldPadExact])

'&*' (fs :: [RngMPolElt_FldPadExact])

-> RngMPolElt_FldPadExact

Negate, add, subtract, multiply, divide by scalar, sum, product.

Parameters

• Safe := false: When true, this may be used as an intermediate variable in WithDependencies with the Fast option.

Derivative

Derivative (f :: RngMPolElt_FldPadExact, m :: RngIntElt, v :: RngIntElt)

Derivative (f :: RngMPolElt_FldPadExact, m :: RngIntElt, v :: RngMPolElt_FldPadExact)

Derivative (f :: RngMPolElt_FldPadExact, v :: RngIntElt)

Derivative (f :: RngMPolElt_FldPadExact, v :: RngMPolElt_FldPadExact)

-> RngMPolElt_FldPadExact

The mth or first derivative of f with respect to variable v.

Evaluate

Evaluate (f :: RngMPolElt_FldPadExact, xs :: [FldPadExactElt])

-> FldPadExactElt

Evaluates f(xs).

Hensel lifting

IsHenselLiftable (fs :: [RngMPolElt_FldPadExact], xs :: [FldPadExactElt])

-> BoolElt, []

True if xs are Hensel liftable to a system of roots of fs. If so, also returns the system of roots.

fs must be a system of n equations of rank n, and xs must be a sequence of n p-adic numbers.

Parameters

• Strategy := "default": The precision strategy to use.
• Slopes: When given, must be a sequence of n rationals to slope the equations by. That is, conceptually we multiply the ith variable by pi^Slopes[i] and xs[i] correspondingly by pi^-Slopes[i]. When not given, the zero slope is used.
• Shifts: When given, must be a sequence of n rationals to shift the equations by. That is, conceptually we multiply the ith equation by pi^Shifts[i]. When not given, the best shifts are chosen.
• AsTuple := false: When true, the solution is returned as a Tup_FldPad, not as a sequence.

Internals

Approximation

WeakMonomials (f :: RngMPolElt_FldPadExact)

-> []

The monomials of f. Some of the corresponding coefficients may be zero.

WeakCoefficients (f :: RngMPolElt_FldPadExact)

-> []

The coefficients of f corresponding to WeakMonomials(f).

WeakCoefficientsAndMonomials (f :: RngMPolElt_FldPadExact)

-> [], []

The coefficients and monomials of f.