Basic Calculator IV

Represent each expression as a map from a sorted monomial key to an integer coefficient. The constant is an empty key; a variable term such as a times b uses the sorted key [a,b]. Parse the expression with precedence: factors are numbers, variables, or parenthesized expressions; products combine factors; sums combine products. Substitute variables found in the evaluation map immediately, otherwise create a one-term polynomial. Addition and subtraction add coefficients by key. Multiplication takes every pair of terms, concatenates and sorts their variable keys, and adds coefficient products; remove entries whose coefficient becomes zero. Finally sort nonconstant terms by decreasing variable count and lexicographic key, then format coefficient followed by variables. Be careful that unary minus and spaces need explicit token handling. Let T be the number of retained terms; multiplication is O(T squared times L) for monomial length L, and memory is O(T times L), with expression size affecting parsing too.