simplify_full returns odd result from symbolic series input

[rwst]

SR.series will lose the order term when passed to Maxima. Thus only the coefficients may be simplified, and this must be done in all simplify* functions.

sage: x=var('x')
sage: s=(1/(1-x)).series(x,6)
sage: s.coeffs()
[[x^5 + x^4 + x^3 + x^2 + x + Order(x^6) + 1, 0]]
sage: s.simplify_full().coeffs()
[[Order(x^6) + 1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5]]

See also the related #17399.

Originally found in http://ask.sagemath.org/question/24968/coefficients-in-polynomial-ring-over-symbolic-ring/

Also, series should simplify its terms on a per-term basis:

sage: var('x,y')
(x, y)
sage: ex=1/(1-x*y-x^2)
sage: ex.series(x,5)
1 + (y)*x + (y^2 + 1)*x^2 + ((y^2 + 1)*y + y)*x^3 + (((y^2 + 1)*y + y)*y + y^2 + 1)*x^4 + Order(x^5)

Compare with e.g. Pari:

? 1/(1-x*y-x^2) + O(x^5)
%1 = 1 + y*x + (y^2 + 1)*x^2 + (y^3 + 2*y)*x^3 + (y^4 + 3*y^2 + 1)*x^4 + O(x^5)

Both issues can be fixed by writing series simplification methods.

Depends on #17399
Depends on #17659

Component: symbolics

Author: Ralf Stephan

Branch/Commit: u/rws/17400-1 @ 22c947a

Issue created by migration from https://trac.sagemath.org/ticket/17400

[rwst]

Description changed:

--- 
+++ 
@@ -8,3 +8,5 @@
 [[Order(x^6) + 1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5]]
 ```
 See also the related #17399.
+
+Originally found in http://ask.sagemath.org/question/24968/coefficients-in-polynomial-ring-over-symbolic-ring/

[rwst]

Description changed:

--- 
+++ 
@@ -7,6 +7,6 @@
 sage: s.simplify_full().coeffs()
 [[Order(x^6) + 1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5]]
 ```
-See also the related #17399.
+See also the related #13655 and #17399.
 
 Originally found in http://ask.sagemath.org/question/24968/coefficients-in-polynomial-ring-over-symbolic-ring/

[rwst]

comment:3

Probably with symbolic series, only coefficients should be simplified in the simplify* functions.

[rwst]

comment:4

There are no power series objects in Maxima, just conversion to infinite sums, i.e. formal power series:

sage: maxima.powerseries(x^2+1/(1-x),x,0)
'sum(_SAGE_VAR_x^i2,i2,0,inf)+_SAGE_VAR_x^2
sage: maxima.powerseries(x^2+1/(1-x),x,0).sage()
x^2 + sum(x^i3, i3, 0, +Infinity)

The Taylor series objects have an order parameter on creation, but this does not get output or translated to Sage:

sage: maxima.taylor(1+x+x^2+x^3,x,0,3)
1+_SAGE_VAR_x+_SAGE_VAR_x^2+_SAGE_VAR_x^3
sage: maxima.taylor(1+x+x^2+x^3,x,0,3).sage()
x^3 + x^2 + x + 1

so there is no way around it that SR.series will lose the order term when passed to Maxima. Thus only the coefficients may be simplified, and this must be done in or called from all simplify* functions.

[rwst]

Description changed:

--- 
+++ 
@@ -1,3 +1,4 @@
+`SR`.series will lose the order term when passed to Maxima. Thus only the coefficients may be simplified, and this must be done in all `simplify*` functions.
 
 ```
 sage: x=var('x')
@@ -7,6 +8,6 @@
 sage: s.simplify_full().coeffs()
 [[Order(x^6) + 1, 0], [1, 1], [1, 2], [1, 3], [1, 4], [1, 5]]
 ```
-See also the related #13655 and #17399.
+See also the related #17399.
 
 Originally found in http://ask.sagemath.org/question/24968/coefficients-in-polynomial-ring-over-symbolic-ring/

[rwst]

comment:5

The following functions call Maxima, have a meaning /wrt series, and therefore need to be modified:
simplify, simplify_trig, simplify_rectform, simplify_rational, simplify_log, expand_log, expand_trig.

[rwst]

Description changed:

--- 
+++ 
@@ -11,3 +11,20 @@
 See also the related #17399.
 
 Originally found in http://ask.sagemath.org/question/24968/coefficients-in-polynomial-ring-over-symbolic-ring/
+
+Also, `series` should simplify its terms on a per-term basis:
+
+```
+sage: var('x,y')
+(x, y)
+sage: ex=1/(1-x*y-x^2)
+sage: ex.series(x,5)
+1 + (y)*x + (y^2 + 1)*x^2 + ((y^2 + 1)*y + y)*x^3 + (((y^2 + 1)*y + y)*y + y^2 + 1)*x^4 + Order(x^5)
+```
+Compare with e.g. Pari:
+
+```
+? 1/(1-x*y-x^2) + O(x^5)
+%1 = 1 + y*x + (y^2 + 1)*x^2 + (y^3 + 2*y)*x^3 + (y^4 + 3*y^2 + 1)*x^4 + O(x^5)
+```
+Both issues can be fixed by writing series simplification methods.

[rwst]

Branch: u/rws/simplify_full_returns_odd_result_from_symbolic_series_input

[rwst]

Commit: 2660235

[rwst]

comment:8

Proof of concept.

---

New commits:

a4d2084	17659: make symbolic series subclass of Expression
ed003f0	16203: make imports more specific to prevent import loops
32ae67c	17399: do not let maxima handle ex.series coefficients
99820cf	17399: roll back previous commit to allow merge of 17428
168b659	Merge branch 'u/rws/coefficients_of_symbolic_expressions_revamp' of trac.sagemath.org:sage into t/17399/fix_coefficients_for_symbolic_series
6cd5286	17399: handle series in ex.coefficients()
0b9914f	17399: remove whitespace changes
2e74911	Merge commit '0b9914f8b6cd4cb0f885f0550bfd375d80495a71' of trac.sagemath.org:sage into t/17400/simplify_full_returns_odd_result_from_symbolic_series_input
2660235	17400: implement series.simplify_full()

[rwst]

Dependencies: #17399, #17659