These "dots" on line charts and area charts are called markers in our library. The marker lives inside the plot object. Defining marker visible false in here will hide all markers from all plots like so

plot:{
  marker:{
    visible:false
  }
  ...
}

You can also define this inside your series object. The series object is an array and each index in this array represent a plot. The series object takes all the same attributes that can go in plot, it is just more specific. What do I mean by more specific? If you have one attribute defined in plot and the same attribute defined in series, the series attribute value will take precedence. Another side effect is if you have an attribute defined in one series object, but not another, it will only be applied to that plot.

Example: https://demos.zingchart.com/view/4IJIZ2PU 

Documentation

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