#5227 (enhancement) one predicate for "a power with an even numerator and an odd denominator"

[David Scherfgen]

Written by Claude.

Nice to have: one predicate for "a power with an even numerator and an odd denominator", so that abs(x)^(2/m) and x^n*abs(x)^(1/3) with a declared parity simplify like the numeric cases, and the places that test it share the code

With domain : real a power x^(p/q) with an even p and an odd q is the real root of x^(1/q) squared, so it is abs(x)^(p/q), and the simplifier uses that: abs(x)^(2/3) is x^(2/3), and x^(2/3)*abs(x)^(1/3) is abs(x). It uses it only when p/q is a number. When the parity is known but symbolic, n declared even or m declared odd, some of it happens and some does not:

(%i1) display2d : false$
(%i2) declare(n, even, m, odd)$
(%i3) [abs(x)^n, abs(x)^(2*m), abs(x)^(2/3), abs(x)^(2/m), abs(x)^(n/3), abs(x)^(2*n/m)];
(%o3) [x^n,x^(2*m),x^(2/3),abs(x)^(2/m),abs(x)^(n/3),abs(x)^((2*n)/m)]
(%i4) [x^(2/3)*abs(x)^(1/3), x^n*abs(x)^(1/3), x^(2/m)*abs(x)^(1/m), x^(n/3)*abs(x)^(1/3), x^4/abs(x), x^n/abs(x)];
(%o4) [abs(x),x^n*abs(x)^(1/3),x^(2/m)*abs(x)^(1/m),x^(n/3)*abs(x)^(1/3),x^2*abs(x),x^n/abs(x)]
(%i5) map(csign, [x^(2/m), x^(n/3), x^(2*n/m), x^(1/n)]);
(%o5) [pz,pz,pz,complex]

csign already knows that x^(2/m) and x^(n/3) are nonnegative, but the abs rules do not: abs(x)^(2/m) stays, x^(2/m)*abs(x)^(1/m) could be abs(x)^(3/m) and x^n*abs(x)^(1/3) could be abs(x)^(n+1/3). Nothing here gives a wrong result; x^(1/n) is an even root, csign says complex, and nothing touches it. It is a missed simplification, and the reason is that the parity of an exponent is tested in four places with four different pieces of code:

• simpexpt in src/simp.lisp, the clause that turns abs(x)^pot into x^pot, tests (mevenp pot), which through evod in src/compar.lisp handles a number, a declared symbol, and products, sums and non-negative integer powers of those, plus (evnump pot) for a rational with an even numerator, with domain : real. evod does not look into a reciprocal or a rational coefficient, so 2/m and n/3 are not recognized.
• The timesin clauses that combine x^a*abs(x)^b into abs(x)^(a+b) test (and (ratnump a) (evenp (cadr a))) for a and (or (integerp b) (ratnump b)) for b: numbers only.
• sign-mexpt in src/compar.lisp judges b^(p/q) by its numerator when (eq (evod q) '$odd), so it does see 2/m and n/3.
• odd-root-p in src/rpart.lisp, used by rectform, carg, the integrator and the derivative of gamma_incomplete, tests (maxima-integerp ($num pow)) and (eq (evod ($denom pow)) '$odd), the same idea again.

The four older abs clauses of timesin, the ones that cancel even powers between abs(x) and x, are a fifth place, with integerp only: x^4/abs(x) is x^2*abs(x), x^n/abs(x) stays.

A single predicate, "an exponent whose numerator is even and whose denominator is odd, by evod on $num and $denom", would answer for 2/3, n, 2*m, 2/m, n/3 and 2*n/m alike, and could replace the parity test in all of these places: in simpexpt next to mevenp under domain : real, in the timesin combination for a, with the integer case still excluded there, and as the numerator half of odd-root-p and of the sign-mexpt rule. That is fewer lines than there are today, one definition to get right instead of four, and the symbolic cases come for free. Only domain : real may use the fraction half of it; with domain : complex x^(2/m) is on the principal branch, where the identity fails for a negative x, and the same restriction that simpexpt already has for 2/3 applies.