The Kardar-Parisi-Zhang exponents for the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si72.svg"><mml:mrow><mml:mn>2</mml:mn><mml:mo linebreak="badbreak">+</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math> dimensions

The Kardar-Parisi-Zhang (KPZ) equation has been connected to a large number of important stochastic processes in physics, chemistry and growth phenomena, ranging from classical quantum physics. central quest this field is the search for ever more precise universal exponents. Notably, exact exponents are only known $1+1$ dimensions. In work, we present physical geometric analytical methods that directly associate these fractal dimension rough interface. Based on this, determine $2+1$ dimensions, which agreement with results thin films experiments simulations. We also make first step towards solution $d+1$ where our suggest inexistence an upper critical dimension.