FriCAS expression domains are based on notion of differential field. This have advantages, for example some otherwise unsolvable problems became solvable when dealing with differential fields. But this also means that FriCAS expression behave differently than function in real analysis. For example, in field quadratic equation have 0, 1 or 2 roots. In particular consider equation where is variable of the equation and is element of differential field representing variable . This equation have exactly two roots, namely and . This means that absolute value of real function is excluded, since for real we have , so would be solution to , but is different than or . In fact, forbiding absolute value is crucial to solvability --- Richardson proved that equality for expressions build from variable , rational constants, constant , trigonometric functions and absolute values by composition and arithmetic operations is undecidable. When we put functions above inside a differential field equality becomes decidable (note: "put inside" means that we provide extra information).

By Seidenberg theorem finitely generated differential field of characteristic is izomorphic to a subfield of field of meromorphic functions in some complex domain (open and connected set). And opposite: field of meromorphic functions in complex domain is a differential field. So differential field correspond very well to complex analysis. But there may be some mismatch when one wants to deal with real analysis. Dealing with differential field we implicitly remove inessential singularities, so we treat and as the same function (for more background see Division by zero during evaluation). Also, by principle of analytic continuation we may restrict functions to arbitrarily small open set and still get izomorphic field. Which means that to get interesting results we may be forced to perform analytic continuation. Since we work with connected sets, we have to chose some branch of function which otherwise is considered as multivalued function. Notably, we have to chose branches of logarithms and roots. Interestingly, in many cases choice of branches does not matter much, we get equivalent results for different choices. What matters are dependencies. For example at algebraic level it does not matter which square roots of and we chose, all four choices give isomorphic fields. However, when we add square root of , we get dependence, either or . In numerical computations it is usual to make choice of root, for example to use so called principal branch of square root. However, in purely algebraic context there is no canonical way to distinguish roots and all roots of the same irreducible equation play equivalent role. In context of differential fields logarithm is determined up to additive constant and there is no way to have canonical choice of constant, all choices lead to izomorphic differential fields.

Important property of Expression(Integer) is that FriCAS does not enforce having differential field. FriCAS allows dependent root, which as shown above lead to zero divisors. FriCAS allows both exp(x)^2 and exp(2*x) in a single expression and by default treats them as different quantities. Note that using differential calculus one can prove that exp(x)^2 and exp(2*x) are equal. This means that when both are used in a single expression then laws of differential field my by violated. FriCAS has abs and diracDelta and few other expressions of similar nature. In arithmetic such expression are treated as transcendental quatities. This is sound in sense that succesfull computation can not produce false equality. But many such expression does not correspond to things in analitic world. FriCAS does not have complete knowledge about such expressions, so in some cases can not discover when in analytic world given expression is 0. Worse, such unrecognized 0 can appear in the denominator. If some later transformation manages to simplify the denominator, it may lead to division by 0.

Computations which depend on having differential field usually call normalize or similar function. This removes various redundancies which for transcendental elementary functions means that computations with transformed expression will work within a differential field. In principle algebraic quantities (like square roots) and abs can be treated by considering possible cases, but currently in few simple cases FriCAS takes single choice and other cases are not handled at all.

Compared to some other systems FriCAS approach have both advantages:

• FriCAS avoids unsolvability due to Richardson theorem
• there are powerful algorithms working in differential fields

but also some disadvantages. Main limitation is that with FriCAS approach it is harder to handle solvable instances of problems involving absolute value.