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15 (* This file was automatically generated: do not edit *********************)
19 (* $Id: CPoly_Degree.v,v 1.5 2004/04/23 10:00:53 lcf Exp $ *)
21 include "algebra/CPoly_NthCoeff.ma".
23 include "algebra/CFields.ma".
25 (*#* *Degrees of Polynomials
26 ** Degrees of polynomials over a ring
28 Let [R] be a ring and write [RX] for the ring of polynomials
37 alias id "R" = "cic:/CoRN/algebra/CPoly_Degree/Degree_def/R.var".
42 Notation RX := (cpoly_cring R).
48 The length of a polynomial is the number of its coefficients. This is
49 a syntactical property, as the highest coefficient may be [0]. Note that
50 the `zero' polynomial [cpoly_zero] has length [0],
51 a constant polynomial has length [1] and so forth. So the length
52 is always [1] higher than the `degree' (assuming that the highest
53 coefficient is [[#]Zero])!
56 inline procedural "cic:/CoRN/algebra/CPoly_Degree/lth_of_poly.con".
59 When dealing with constructive polynomials, notably over the reals or
60 complex numbers, the degree may be unknown, as we can not decide
61 whether the highest coefficient is [[#]Zero]. Hence,
62 degree is a relation between polynomials and natural numbers; if the
63 degree is unknown for polynomial [p], degree(n,p) doesn't hold for
64 any [n]. If we don't know the degree of [p], we may still
65 know it to be below or above a certain number. E.g. for the polynomial
66 $p_0 +p_1 X +\cdots + p_{n-1} X^{n-1}$#p0 +p1 X + ... + p(n-1)
67 X^(n-1)#, if $p_i \mathrel{\#}0$#pi apart from 0#, we can say that the
68 `degree is at least [i]' and if $p_{j+1} = \ldots =p_n =0$#p(j+1)
69 = ... =pn =0# (with [n] the length of the polynomial), we can say
70 that the `degree is at most [j]'.
73 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le.con".
75 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree.con".
77 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic.con".
79 inline procedural "cic:/CoRN/algebra/CPoly_Degree/odd_cpoly.con".
81 inline procedural "cic:/CoRN/algebra/CPoly_Degree/even_cpoly.con".
83 inline procedural "cic:/CoRN/algebra/CPoly_Degree/regular.con".
90 Implicit Arguments degree_le [R].
94 Implicit Arguments degree [R].
98 Implicit Arguments monic [R].
102 Implicit Arguments lth_of_poly [R].
109 alias id "R" = "cic:/CoRN/algebra/CPoly_Degree/Degree_props/R.var".
114 Notation RX := (cpoly_cring R).
119 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_wd.con".
121 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_wd.con".
123 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_wd.con".
125 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_imp_degree_le.con".
127 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_c_.con".
129 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_c_.con".
131 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_c_one.con".
133 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_x_.con".
135 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_x_.con".
137 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_x_.con".
139 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_mon.con".
141 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_inv.con".
143 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_plus.con".
145 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_minus.con".
147 inline procedural "cic:/CoRN/algebra/CPoly_Degree/Sum_degree_le.con".
149 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_inv.con".
151 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_plus_rht.con".
153 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_minus_lft.con".
155 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_plus.con".
157 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_minus.con".
159 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_mult.con".
161 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_mult_aux.con".
164 Hint Resolve degree_mult_aux: algebra.
167 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_mult.con".
169 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_nexp.con".
171 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_nexp.con".
173 inline procedural "cic:/CoRN/algebra/CPoly_Degree/lt_i_lth_of_poly.con".
175 inline procedural "cic:/CoRN/algebra/CPoly_Degree/poly_degree_lth.con".
177 inline procedural "cic:/CoRN/algebra/CPoly_Degree/Cpoly_ex_degree.con".
179 inline procedural "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum''.con".
182 Hint Resolve poly_as_sum'': algebra.
185 inline procedural "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum'.con".
187 inline procedural "cic:/CoRN/algebra/CPoly_Degree/poly_as_sum.con".
189 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_zero.con".
191 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_1_imp.con".
193 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_cpoly_linear.con".
195 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_cpoly_linear.con".
197 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_one.con".
199 inline procedural "cic:/CoRN/algebra/CPoly_Degree/monic_apzero.con".
206 Hint Resolve poly_as_sum'' poly_as_sum' poly_as_sum: algebra.
210 Hint Resolve degree_mult_aux: algebra.
214 Section degree_props_Field
217 (*#* ** Degrees of polynomials over a field
218 %\begin{convention}% Let [F] be a field and write [FX] for the ring of
219 polynomials over [F].
223 alias id "F" = "cic:/CoRN/algebra/CPoly_Degree/degree_props_Field/F.var".
228 Notation FX := (cpoly_cring F).
233 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_mult.con".
235 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_nexp.con".
237 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_le_mult_imp.con".
239 inline procedural "cic:/CoRN/algebra/CPoly_Degree/degree_mult_imp.con".
242 End degree_props_Field