# Proper way to write a regression equation with multiple interaction terms

As I am about to submit my master thesis, I am having doubts about the mathematical correctness of the regression equation I have used to represent my model. A bit of context: I am predicting review rating with 12 continuous variables (labelled topics 1 to 12) and an industry factor (that can be "Finance", "Tech", "FMCG" and "Accounting"). On top of that, there is an interaction term for each of the combinations between the 12 continuous variables and the industries (variable 1 with finance, variable 1 with tech, variable 1 with FMCG, variable 1 with accounting and so on...). I wonder if there is a simplified way of representing this in the equation as otherwise, the equation has more than 50 coefficients which makes it not neat.

This is what I want to simplify (note that this is an example with just two continuous variables and two industries so that it doesn't take too much space and time):

$$\widehat{Rating} = \widehat{\beta_0} + {\widehat{\beta_{Topic1}}X_{Topic1}} + {\widehat{\beta_{Topic2}}X_{Topic2}} +{\widehat{\beta_{TechIndustry}}X_{TechIndustry}}+ {\widehat{\beta_{FinanceIndustry}}X_{FinanceIndustry}}+{\widehat{\beta_{Topic1*TechIndustry}}X_{Topic1*TechIndustry}}+{\widehat{\beta_{Topic1*FinanceIndustry}}X_{Topic1*FinanceIndustry}}+{\widehat{\beta_{Topic2*TechIndustry}}X_{Topic2*TechIndustry}}+{\widehat{\beta_{Topic2*FinanceIndustry}}X_{Topic2*FinanceIndustry}}+\epsilon$$

How can I express this function in a simplified way (maybe with indexes)??